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Name":1149},[75,162],[19,1151,46],{"className":1152},[81],[19,1154,1156,1159],{"className":1155},[75],[19,1157,49],{"className":1158},[75,76],[19,1160,1162],{"className":1161},[133],[19,1163,1165,1188],{"className":1164},[137,138],[19,1166,1168,1185],{"className":1167},[142],[19,1169,1171],{"className":1170,"style":147},[146],[19,1172,1173,1176],{"style":150},[19,1174],{"className":1175,"style":155},[154],[19,1177,1179],{"className":1178},[159,160,161,162],[19,1180,1182],{"className":1181},[75,162],[19,1183,110],{"className":1184},[75,162],[19,1186,173],{"className":1187},[172],[19,1189,1191],{"className":1190},[142],[19,1192,1194],{"className":1193,"style":180},[146],[19,1195],{},[19,1197,52],{"className":1198},[88],[19,1200],{"className":1201,"style":613},[411],[19,1203,267],{"className":1204},[335],[19,1206],{"className":1207,"style":613},[411],[19,1209,1211,1215],{"className":1210},[66],[19,1212],{"className":1213,"style":1214},[70],"height:0.6444em;",[19,1216,110],{"className":1217},[75],[15,1219,1220],{},"Rearranging gives",[19,1222,1224],{"className":1223,"translate":23},[187],[19,1225,1227,1333],{"className":1226,"translate":23},[22],[19,1228,1230],{"className":1229},[27],[29,1231,1232],{"xmlns":31,"display":196},[33,1233,1234,1330],{},[36,1235,1236,1256,1326,1328],{},[202,1237,1238,1244],{},[36,1239,1240,1242],{},[39,1241,208],{},[43,1243,211],{},[36,1245,1246,1248,1250],{},[39,1247,49],{},[43,1249,218],{},[103,1251,1252,1254],{},[39,1253,49],{},[108,1255,110],{},[226,1257,1258,1314],{},[36,1259,1260,1262,1264,1266,1268,1270,1272,1274,1280,1282,1284,1290,1292,1298,1300,1302,1304,1306,1312],{},[39,1261,41],{},[43,1263,46],{"stretchy":45},[39,1265,49],{},[43,1267,52],{"stretchy":45},[43,1269,240],{},[39,1271,41],{},[43,1273,46],{"stretchy":45},[103,1275,1276,1278],{},[39,1277,49],{},[108,1279,110],{},[43,1281,52],{"stretchy":45},[43,1283,240],{},[269,1285,1286,1288],{},[39,1287,41],{},[43,1289,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,1291,46],{"stretchy":45},[103,1293,1294,1296],{},[39,1295,49],{},[108,1297,110],{},[43,1299,52],{"stretchy":45},[43,1301,46],{"stretchy":45},[39,1303,49],{},[43,1305,240],{},[103,1307,1308,1310],{},[39,1309,49],{},[108,1311,110],{},[43,1313,52],{"stretchy":45},[36,1315,1316,1318,1320],{},[39,1317,49],{},[43,1319,240],{},[103,1321,1322,1324],{},[39,1323,49],{},[108,1325,110],{},[43,1327,267],{},[108,1329,110],{},[54,1331,1332],{"encoding":56},"\\lim_{x \\rightarrow 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terms 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x_{0})",[19,2418,2420,2447,2514,2616,2677,2701,2768],{"className":2419,"ariaHidden":62},[61],[19,2421,2423,2426,2429,2432,2435,2438,2441,2444],{"className":2422},[66],[19,2424],{"className":2425,"style":71},[70],[19,2427,41],{"className":2428,"style":77},[75,76],[19,2430,46],{"className":2431},[81],[19,2433,49],{"className":2434},[75,76],[19,2436,52],{"className":2437},[88],[19,2439],{"className":2440,"style":613},[411],[19,2442,267],{"className":2443},[335],[19,2445],{"className":2446,"style":613},[411],[19,2448,2450,2453,2456,2459,2502,2505,2508,2511],{"className":2449},[66],[19,2451],{"className":2452,"style":71},[70],[19,2454,41],{"className":2455,"style":77},[75,76],[19,2457,46],{"className":2458},[81],[19,2460,2462,2465],{"className":2461},[75],[19,2463,49],{"className":2464},[75,76],[19,2466,2468],{"className":2467},[133],[19,2469,2471,2494],{"className":2470},[137,138],[19,2472,2474,2491],{"className":2473},[142],[19,2475,2477],{"className":2476,"style":147},[146],[19,2478,2479,2482],{"style":150},[19,2480],{"className":2481,"style":155},[154],[19,2483,2485],{"className":2484},[159,160,161,162],[19,2486,2488],{"className":2487},[75,162],[19,2489,110],{"className":2490},[75,162],[19,2492,173],{"className":2493},[172],[19,2495,2497],{"className":2496},[142],[19,2498,2500],{"className":2499,"style":180},[146],[19,2501],{},[19,2503,52],{"className":2504},[88],[19,2506],{"className":2507,"style":451},[411],[19,2509,2351],{"className":2510},[455],[19,2512],{"className":2513,"style":451},[411],[19,2515,2517,2520,2552,2555,2598,2601,2604,2607,2610,2613],{"className":2516},[66],[19,2518],{"className":2519,"style":626},[70],[19,2521,2523,2526],{"className":2522},[75],[19,2524,41],{"className":2525,"style":77},[75,76],[19,2527,2529],{"className":2528},[133],[19,2530,2532],{"className":2531},[137],[19,2533,2535],{"className":2534},[142],[19,2536,2538],{"className":2537,"style":645},[146],[19,2539,2540,2543],{"style":648},[19,2541],{"className":2542,"style":155},[154],[19,2544,2546],{"className":2545},[159,160,161,162],[19,2547,2549],{"className":2548},[75,162],[19,2550,277],{"className":2551},[75,162],[19,2553,46],{"className":2554},[81],[19,2556,2558,2561],{"className":2557},[75],[19,2559,49],{"className":2560},[75,76],[19,2562,2564],{"className":2563},[133],[19,2565,2567,2590],{"className":2566},[137,138],[19,2568,2570,2587],{"className":2569},[142],[19,2571,2573],{"className":2572,"style":147},[146],[19,2574,2575,2578],{"style":150},[19,2576],{"className":2577,"style":155},[154],[19,2579,2581],{"className":2580},[159,160,161,162],[19,2582,2584],{"className":2583},[75,162],[19,2585,110],{"className":2586},[75,162],[19,2588,173],{"className":2589},[172],[19,2591,2593],{"className":2592},[142],[19,2594,2596],{"className":2595,"style":180},[146],[19,2597],{},[19,2599,52],{"className":2600},[88],[19,2602,46],{"className":2603},[81],[19,2605,49],{"className":2606},[75,76],[19,2608],{"className":2609,"style":451},[411],[19,2611,240],{"className":2612},[455],[19,2614],{"className":2615,"style":451},[411],[19,2617,2619,2622,2665,2668,2671,2674],{"className":2618},[66],[19,2620],{"className":2621,"style":71},[70],[19,2623,2625,2628],{"className":2624},[75],[19,2626,49],{"className":2627},[75,76],[19,2629,2631],{"className":2630},[133],[19,2632,2634,2657],{"className":2633},[137,138],[19,2635,2637,2654],{"className":2636},[142],[19,2638,2640],{"className":2639,"style":147},[146],[19,2641,2642,2645],{"style":150},[19,2643],{"className":2644,"style":155},[154],[19,2646,2648],{"className":2647},[159,160,161,162],[19,2649,2651],{"className":2650},[75,162],[19,2652,110],{"className":2653},[75,162],[19,2655,173],{"className":2656},[172],[19,2658,2660],{"className":2659},[142],[19,2661,2663],{"className":2662,"style":180},[146],[19,2664],{},[19,2666,52],{"className":2667},[88],[19,2669],{"className":2670,"style":451},[411],[19,2672,2351],{"className":2673},[455],[19,2675],{"className":2676,"style":451},[411],[19,2678,2680,2683,2686,2689,2692,2695,2698],{"className":2679},[66],[19,2681],{"className":2682,"style":71},[70],[19,2684,1874],{"className":2685},[75,76],[19,2687,46],{"className":2688},[81],[19,2690,49],{"className":2691},[75,76],[19,2693],{"className":2694,"style":451},[411],[19,2696,240],{"className":2697},[455],[19,2699],{"className":2700,"style":451},[411],[19,2702,2704,2707,2750,2753,2756,2759,2762,2765],{"className":2703},[66],[19,2705],{"className":2706,"style":71},[70],[19,2708,2710,2713],{"className":2709},[75],[19,2711,49],{"className":2712},[75,76],[19,2714,2716],{"className":2715},[133],[19,2717,2719,2742],{"className":2718},[137,138],[19,2720,2722,2739],{"className":2721},[142],[19,2723,2725],{"className":2724,"style":147},[146],[19,2726,2727,2730],{"style":150},[19,2728],{"className":2729,"style":155},[154],[19,2731,2733],{"className":2732},[159,160,161,162],[19,2734,2736],{"className":2735},[75,162],[19,2737,110],{"className":2738},[75,162],[19,2740,173],{"className":2741},[172],[19,2743,2745],{"className":2744},[142],[19,2746,2748],{"className":2747,"style":180},[146],[19,2749],{},[19,2751,52],{"className":2752},[88],[19,2754,46],{"className":2755},[81],[19,2757,49],{"className":2758},[75,76],[19,2760],{"className":2761,"style":613},[411],[19,2763,218],{"className":2764},[335],[19,2766],{"className":2767,"style":613},[411],[19,2769,2771,2774,2817],{"className":2770},[66],[19,2772],{"className":2773,"style":71},[70],[19,2775,2777,2780],{"className":2776},[75],[19,2778,49],{"className":2779},[75,76],[19,2781,2783],{"className":2782},[133],[19,2784,2786,2809],{"className":2785},[137,138],[19,2787,2789,2806],{"className":2788},[142],[19,2790,2792],{"className":2791,"style":147},[146],[19,2793,2794,2797],{"style":150},[19,2795],{"className":2796,"style":155},[154],[19,2798,2800],{"className":2799},[159,160,161,162],[19,2801,2803],{"className":2802},[75,162],[19,2804,110],{"className":2805},[75,162],[19,2807,173],{"className":2808},[172],[19,2810,2812],{"className":2811},[142],[19,2813,2815],{"className":2814,"style":180},[146],[19,2816],{},[19,2818,52],{"className":2819},[88],[15,2821,2822,2823,2879,2880,2923,2924,3151,3152,3238,3239,3282,3283,3313,3314,3447],{},"This implies that near the point ",[19,2824,2826,2845],{"className":2825,"translate":23},[22],[19,2827,2829],{"className":2828},[27],[29,2830,2831],{"xmlns":31},[33,2832,2833,2842],{},[36,2834,2835,2837,2840],{},[39,2836,49],{},[43,2838,2839],{},"∗",[108,2841,110],{},[54,2843,2844],{"encoding":56},"x*{0}",[19,2846,2848,2867],{"className":2847,"ariaHidden":62},[61],[19,2849,2851,2855,2858,2861,2864],{"className":2850},[66],[19,2852],{"className":2853,"style":2854},[70],"height:0.4653em;",[19,2856,49],{"className":2857},[75,76],[19,2859],{"className":2860,"style":451},[411],[19,2862,2839],{"className":2863},[455],[19,2865],{"className":2866,"style":451},[411],[19,2868,2870,2873],{"className":2869},[66],[19,2871],{"className":2872,"style":1214},[70],[19,2874,2876],{"className":2875},[75],[19,2877,110],{"className":2878},[75],", we can approximate the function ",[19,2881,2883,2902],{"className":2882,"translate":23},[22],[19,2884,2886],{"className":2885},[27],[29,2887,2888],{"xmlns":31},[33,2889,2890,2900],{},[36,2891,2892,2894,2896,2898],{},[39,2893,41],{},[43,2895,46],{"stretchy":45},[39,2897,49],{},[43,2899,52],{"stretchy":45},[54,2901,57],{"encoding":56},[19,2903,2905],{"className":2904,"ariaHidden":62},[61],[19,2906,2908,2911,2914,2917,2920],{"className":2907},[66],[19,2909],{"className":2910,"style":71},[70],[19,2912,41],{"className":2913,"style":77},[75,76],[19,2915,46],{"className":2916},[81],[19,2918,49],{"className":2919},[75,76],[19,2921,52],{"className":2922},[88]," with a first-degree polynomial ",[19,2925,2927,2983],{"className":2926,"translate":23},[22],[19,2928,2930],{"className":2929},[27],[29,2931,2932],{"xmlns":31},[33,2933,2934,2980],{},[36,2935,2936,2938,2940,2942,2944,2946,2948,2950,2956,2958,2960,2962,2964,2966,2968,2970,2972,2974,2976,2978],{},[39,2937,41],{},[43,2939,46],{"stretchy":45},[39,2941,49],{},[43,2943,2839],{},[108,2945,110],{},[43,2947,52],{"stretchy":45},[43,2949,2351],{},[269,2951,2952,2954],{},[39,2953,41],{},[43,2955,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,2957,46],{"stretchy":45},[39,2959,49],{},[43,2961,2839],{},[108,2963,110],{},[43,2965,52],{"stretchy":45},[43,2967,46],{"stretchy":45},[39,2969,49],{},[43,2971,240],{},[39,2973,49],{},[43,2975,2839],{},[108,2977,110],{},[43,2979,52],{"stretchy":45},[54,2981,2982],{"encoding":56},"f(x*{0})+f'(x*{0})(x-x*{0})",[19,2984,2986,3010,3034,3088,3118,3136],{"className":2985,"ariaHidden":62},[61],[19,2987,2989,2992,2995,2998,3001,3004,3007],{"className":2988},[66],[19,2990],{"className":2991,"style":71},[70],[19,2993,41],{"className":2994,"style":77},[75,76],[19,2996,46],{"className":2997},[81],[19,2999,49],{"className":3000},[75,76],[19,3002],{"className":3003,"style":451},[411],[19,3005,2839],{"className":3006},[455],[19,3008],{"className":3009,"style":451},[411],[19,3011,3013,3016,3022,3025,3028,3031],{"className":3012},[66],[19,3014],{"className":3015,"style":71},[70],[19,3017,3019],{"className":3018},[75],[19,3020,110],{"className":3021},[75],[19,3023,52],{"className":3024},[88],[19,3026],{"className":3027,"style":451},[411],[19,3029,2351],{"className":3030},[455],[19,3032],{"className":3033,"style":451},[411],[19,3035,3037,3041,3073,3076,3079,3082,3085],{"className":3036},[66],[19,3038],{"className":3039,"style":3040},[70],"height:1.0019em;vertical-align:-0.25em;",[19,3042,3044,3047],{"className":3043},[75],[19,3045,41],{"className":3046,"style":77},[75,76],[19,3048,3050],{"className":3049},[133],[19,3051,3053],{"className":3052},[137],[19,3054,3056],{"className":3055},[142],[19,3057,3059],{"className":3058,"style":1640},[146],[19,3060,3061,3064],{"style":1643},[19,3062],{"className":3063,"style":155},[154],[19,3065,3067],{"className":3066},[159,160,161,162],[19,3068,3070],{"className":3069},[75,162],[19,3071,277],{"className":3072},[75,162],[19,3074,46],{"className":3075},[81],[19,3077,49],{"className":3078},[75,76],[19,3080],{"className":3081,"style":451},[411],[19,3083,2839],{"className":3084},[455],[19,3086],{"className":3087,"style":451},[411],[19,3089,3091,3094,3100,3103,3106,3109,3112,3115],{"className":3090},[66],[19,3092],{"className":3093,"style":71},[70],[19,3095,3097],{"className":3096},[75],[19,3098,110],{"className":3099},[75],[19,3101,52],{"className":3102},[88],[19,3104,46],{"className":3105},[81],[19,3107,49],{"className":3108},[75,76],[19,3110],{"className":3111,"style":451},[411],[19,3113,240],{"className":3114},[455],[19,3116],{"className":3117,"style":451},[411],[19,3119,3121,3124,3127,3130,3133],{"className":3120},[66],[19,3122],{"className":3123,"style":2854},[70],[19,3125,49],{"className":3126},[75,76],[19,3128],{"className":3129,"style":451},[411],[19,3131,2839],{"className":3132},[455],[19,3134],{"className":3135,"style":451},[411],[19,3137,3139,3142,3148],{"className":3138},[66],[19,3140],{"className":3141,"style":71},[70],[19,3143,3145],{"className":3144},[75],[19,3146,110],{"className":3147},[75],[19,3149,52],{"className":3150},[88],", with an error of ",[19,3153,3155,3181],{"className":3154,"translate":23},[22],[19,3156,3158],{"className":3157},[27],[29,3159,3160],{"xmlns":31},[33,3161,3162,3178],{},[36,3163,3164,3166,3168,3170,3172,3174,3176],{},[43,3165,46],{"stretchy":45},[39,3167,49],{},[43,3169,240],{},[39,3171,49],{},[43,3173,2839],{},[108,3175,110],{},[43,3177,52],{"stretchy":45},[54,3179,3180],{"encoding":56},"(x-x*{0})",[19,3182,3184,3205,3223],{"className":3183,"ariaHidden":62},[61],[19,3185,3187,3190,3193,3196,3199,3202],{"className":3186},[66],[19,3188],{"className":3189,"style":71},[70],[19,3191,46],{"className":3192},[81],[19,3194,49],{"className":3195},[75,76],[19,3197],{"className":3198,"style":451},[411],[19,3200,240],{"className":3201},[455],[19,3203],{"className":3204,"style":451},[411],[19,3206,3208,3211,3214,3217,3220],{"className":3207},[66],[19,3209],{"className":3210,"style":2854},[70],[19,3212,49],{"className":3213},[75,76],[19,3215],{"className":3216,"style":451},[411],[19,3218,2839],{"className":3219},[455],[19,3221],{"className":3222,"style":451},[411],[19,3224,3226,3229,3235],{"className":3225},[66],[19,3227],{"className":3228,"style":71},[70],[19,3230,3232],{"className":3231},[75],[19,3233,110],{"className":3234},[75],[19,3236,52],{"className":3237},[88]," as a higher-order infinitesimal. Sometimes, such an approximation may be crude, indicating a relatively large error. Thus, naturally, we wonder if we can approximate ",[19,3240,3242,3261],{"className":3241,"translate":23},[22],[19,3243,3245],{"className":3244},[27],[29,3246,3247],{"xmlns":31},[33,3248,3249,3259],{},[36,3250,3251,3253,3255,3257],{},[39,3252,41],{},[43,3254,46],{"stretchy":45},[39,3256,49],{},[43,3258,52],{"stretchy":45},[54,3260,57],{"encoding":56},[19,3262,3264],{"className":3263,"ariaHidden":62},[61],[19,3265,3267,3270,3273,3276,3279],{"className":3266},[66],[19,3268],{"className":3269,"style":71},[70],[19,3271,41],{"className":3272,"style":77},[75,76],[19,3274,46],{"className":3275},[81],[19,3277,49],{"className":3278},[75,76],[19,3280,52],{"className":3281},[88]," using a higher-degree ",[19,3284,3286,3300],{"className":3285,"translate":23},[22],[19,3287,3289],{"className":3288},[27],[29,3290,3291],{"xmlns":31},[33,3292,3293,3298],{},[36,3294,3295],{},[39,3296,3297],{},"n",[54,3299,3297],{"encoding":56},[19,3301,3303],{"className":3302,"ariaHidden":62},[61],[19,3304,3306,3310],{"className":3305},[66],[19,3307],{"className":3308,"style":3309},[70],"height:0.4306em;",[19,3311,3297],{"className":3312},[75,76]," polynomial to make the error 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a polynomial 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 \\\\\na_{1}=\\frac{f_{n}'(x_{0})}{1!} \\\\\na_{2}=\\frac{f_{n}''(x_{0})}{2!} 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the coefficients of the polynomial function 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are uniquely determined by its derivatives at point ",[19,6238,6240,6257],{"className":6239,"translate":23},[22],[19,6241,6243],{"className":6242},[27],[29,6244,6245],{"xmlns":31},[33,6246,6247,6255],{},[36,6248,6249,6251,6253],{},[39,6250,49],{},[43,6252,2839],{},[108,6254,110],{},[54,6256,2844],{"encoding":56},[19,6258,6260,6278],{"className":6259,"ariaHidden":62},[61],[19,6261,6263,6266,6269,6272,6275],{"className":6262},[66],[19,6264],{"className":6265,"style":2854},[70],[19,6267,49],{"className":6268},[75,76],[19,6270],{"className":6271,"style":451},[411],[19,6273,2839],{"className":6274},[455],[19,6276],{"className":6277,"style":451},[411],[19,6279,6281,6284],{"className":6280},[66],[19,6282],{"className":6283,"style":1214},[70],[19,6285,6287],{"className":6286},[75],[19,6288,110],{"className":6289},[75],". This insight inspires us that for a general function ",[19,6292,6294,6313],{"className":6293,"translate":23},[22],[19,6295,6297],{"className":6296},[27],[29,6298,6299],{"xmlns":31},[33,6300,6301,6311],{},[36,6302,6303,6305,6307,6309],{},[39,6304,41],{},[43,6306,46],{"stretchy":45},[39,6308,49],{},[43,6310,52],{"stretchy":45},[54,6312,57],{"encoding":56},[19,6314,6316],{"className":6315,"ariaHidden":62},[61],[19,6317,6319,6322,6325,6328,6331],{"className":6318},[66],[19,6320],{"className":6321,"style":71},[70],[19,6323,41],{"className":6324,"style":77},[75,76],[19,6326,46],{"className":6327},[81],[19,6329,49],{"className":6330},[75,76],[19,6332,52],{"className":6333},[88],", if 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then these derivatives uniquely determine an ",[19,6482,6484,6497],{"className":6483,"translate":23},[22],[19,6485,6487],{"className":6486},[27],[29,6488,6489],{"xmlns":31},[33,6490,6491,6495],{},[36,6492,6493],{},[39,6494,3297],{},[54,6496,3297],{"encoding":56},[19,6498,6500],{"className":6499,"ariaHidden":62},[61],[19,6501,6503,6506],{"className":6502},[66],[19,6504],{"className":6505,"style":3309},[70],[19,6507,3297],{"className":6508},[75,76],"th degree 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e":8032},[75],[19,8034,7830],{"className":8035},[8036],"mpunct",[19,8038],{"className":8039,"style":412},[411],[19,8041,3518],{"className":8042},[75],[19,8044,7830],{"className":8045},[8036],[19,8047],{"className":8048,"style":412},[411],[19,8050,4007],{"className":8051},[75],[19,8053,7830],{"className":8054},[8036],[19,8056],{"className":8057,"style":412},[411],[19,8059,3297],{"className":8060},[75,76],[19,8062,52],{"className":8063},[88],[15,8065,8066,8067,3447],{},"are termed as ",[7600,8068,8069],{},"Taylor coefficients",[15,8071,8072,8073,8116,8117,8185,8186,6407,8214,183],{},"It is evident that the function ",[19,8074,8076,8095],{"className":8075,"translate":23},[22],[19,8077,8079],{"className":8078},[27],[29,8080,8081],{"xmlns":31},[33,8082,8083,8093],{},[36,8084,8085,8087,8089,8091],{},[39,8086,41],{},[43,8088,46],{"stretchy":45},[39,8090,49],{},[43,8092,52],{"stretchy":45},[54,8094,57],{"encoding":56},[19,8096,8098],{"className":8097,"ariaHidden":62},[61],[19,8099,8101,8104,8107,8110,8113],{"className":8100},[66],[19,8102],{"className":8103,"style":71},[70],[19,8105,41],{"className":8106,"style":77},[75,76],[19,8108,46],{"className":8109},[81],[19,8111,49],{"className":8112},[75,76],[19,8114,52],{"className":8115},[88]," and its Taylor polynomial ",[19,8118,8120,8143],{"className":8119,"translate":23},[22],[19,8121,8123],{"className":8122},[27],[29,8124,8125],{"xmlns":31},[33,8126,8127,8141],{},[36,8128,8129,8131,8133,8135,8137,8139],{},[39,8130,6529],{},[43,8132,2839],{},[39,8134,3297],{},[43,8136,46],{"stretchy":45},[39,8138,49],{},[43,8140,52],{"stretchy":45},[54,8142,7728],{"encoding":56},[19,8144,8146,8164],{"className":8145,"ariaHidden":62},[61],[19,8147,8149,8152,8155,8158,8161],{"className":8148},[66],[19,8150],{"className":8151,"style":7738},[70],[19,8153,6529],{"className":8154,"style":6724},[75,76],[19,8156],{"className":8157,"style":451},[411],[19,8159,2839],{"className":8160},[455],[19,8162],{"className":8163,"style":451},[411],[19,8165,8167,8170,8176,8179,8182],{"className":8166},[66],[19,8168],{"className":8169,"style":71},[70],[19,8171,8173],{"className":8172},[75],[19,8174,3297],{"className":8175},[75,76],[19,8177,46],{"className":8178},[81],[19,8180,49],{"className":8181},[75,76],[19,8183,52],{"className":8184},[88]," have the same function values and derivatives up to the ",[19,8187,8189,8202],{"className":8188,"translate":23},[22],[19,8190,8192],{"className":8191},[27],[29,8193,8194],{"xmlns":31},[33,8195,8196,8200],{},[36,8197,8198],{},[39,8199,3297],{},[54,8201,3297],{"encoding":56},[19,8203,8205],{"className":8204,"ariaHidden":62},[61],[19,8206,8208,8211],{"className":8207},[66],[19,8209],{"className":8210,"style":3309},[70],[19,8212,3297],{"className":8213},[75,76],[19,8215,8217,8234],{"className":8216,"translate":23},[22],[19,8218,8220],{"className":8219},[27],[29,8221,8222],{"xmlns":31},[33,8223,8224,8232],{},[36,8225,8226,8228,8230],{},[39,8227,49],{},[43,8229,2839],{},[108,8231,110],{},[54,8233,2844],{"encoding":56},[19,8235,8237,8255],{"className":8236,"ariaHidden":62},[61],[19,8238,8240,8243,8246,8249,8252],{"className":8239},[66],[19,8241],{"className":8242,"style":2854},[70],[19,8244,49],{"className":8245},[75,76],[19,8247],{"className":8248,"style":451},[411],[19,8250,2839],{"className":8251},[455],[19,8253],{"className":8254,"style":451},[411],[19,8256,8258,8261],{"className":8257},[66],[19,8259],{"className":8260,"style":1214},[70],[19,8262,8264],{"className":8263},[75],[19,8265,110],{"className":8266},[75],[19,8268,8270],{"className":8269,"translate":23},[187],[19,8271,8273,8367],{"className":8272,"translate":23},[22],[19,8274,8276],{"className":8275},[27],[29,8277,8278],{"xmlns":31,"display":196},[33,8279,8280,8364],{},[36,8281,8282,8294,8296,8302,8304,8306,8320,8322,8328,8330,8332,8336,8338,8340,8342,8344,8346,8348,8350,8352,8354,8356,8358,8360,8362],{},[269,8283,8284,8286],{},[39,8285,41],{},[36,8287,8288,8290,8292],{},[43,8289,46],{"stretchy":45},[39,8291,7801],{},[43,8293,52],{"stretchy":45},[43,8295,46],{"stretchy":45},[103,8297,8298,8300],{},[39,8299,49],{},[108,8301,110],{},[43,8303,52],{"stretchy":45},[43,8305,267],{},[4285,8307,8308,8310,8312],{},[39,8309,6529],{},[39,8311,3297],{},[36,8313,8314,8316,8318],{},[43,8315,46],{"stretchy":45},[39,8317,7801],{},[43,8319,52],{"stretchy":45},[43,8321,46],{"stretchy":45},[103,8323,8324,8326],{},[39,8325,49],{},[108,8327,110],{},[43,8329,52],{"stretchy":45},[43,8331,7830],{"separator":62},[8333,8334,8335],"mtext",{}," ",[39,8337,7801],{},[43,8339,267],{},[108,8341,110],{},[43,8343,7830],{"separator":62},[108,8345,3495],{},[43,8347,7830],{"separator":62},[108,8349,3518],{},[43,8351,7830],{"separator":62},[39,8353,3447],{"mathvariant":275},[39,8355,3447],{"mathvariant":275},[39,8357,3447],{"mathvariant":275},[43,8359,7830],{"separator":62},[39,8361,3297],{},[39,8363,3447],{"mathvariant":275},[54,8365,8366],{"encoding":56},"f^{(k)}(x_{0})=T_{n}^{(k)}(x_{0}),\\ k=0,1,2,...,n.",[19,8368,8370,8472,8612],{"className":8369,"ariaHidden":62},[61],[19,8371,8373,8376,8414,8417,8460,8463,8466,8469],{"className":8372},[66],[19,8374],{"className":8375,"style":4934},[70],[19,8377,8379,8382],{"className":8378},[75],[19,8380,41],{"className":8381,"style":77},[75,76],[19,8383,8385],{"className":8384},[133],[19,8386,8388],{"className":8387},[137],[19,8389,8391],{"className":8390},[142],[19,8392,8394],{"className":8393,"style":4953},[146],[19,8395,8396,8399],{"style":648},[19,8397],{"className":8398,"style":155},[154],[19,8400,8402],{"className":8401},[159,160,161,162],[19,8403,8405,8408,8411],{"className":8404},[75,162],[19,8406,46],{"className":8407},[81,162],[19,8409,7801],{"className":8410,"style":7888},[75,76,162],[19,8412,52],{"className":8413},[88,162],[19,8415,46],{"className":8416},[81],[19,8418,8420,8423],{"className":8419},[75],[19,8421,49],{"className":8422},[75,76],[19,8424,8426],{"className":8425},[133],[19,8427,8429,8452],{"className":8428},[137,138],[19,8430,8432,8449],{"className":8431},[142],[19,8433,8435],{"className":8434,"style":147},[146],[19,8436,8437,8440],{"style":150},[19,8438],{"className":8439,"style":155},[154],[19,8441,8443],{"className":8442},[159,160,161,162],[19,8444,8446],{"className":8445},[75,162],[19,8447,110],{"className":8448},[75,162],[19,8450,173],{"className":8451},[172],[19,8453,8455],{"className":8454},[142],[19,8456,8458],{"className":8457,"style":180},[146],[19,8459],{},[19,8461,52],{"className":8462},[88],[19,8464],{"className":8465,"style":613},[411],[19,8467,267],{"className":8468},[335],[19,8470],{"className":8471,"style":613},[411],[19,8473,8475,8478,8542,8545,8588,8591,8594,8597,8600,8603,8606,8609],{"className":8474},[66],[19,8476],{"className":8477,"style":4934},[70],[19,8479,8481,8484],{"className":8480},[75],[19,8482,6529],{"className":8483,"style":6724},[75,76],[19,8485,8487],{"className":8486},[133],[19,8488,8490,8534],{"className":8489},[137,138],[19,8491,8493,8531],{"className":8492},[142],[19,8494,8496,8511],{"className":8495,"style":4953},[146],[19,8497,8499,8502],{"style":8498},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[19,8500],{"className":8501,"style":155},[154],[19,8503,8505],{"className":8504},[159,160,161,162],[19,8506,8508],{"className":8507},[75,162],[19,8509,3297],{"className":8510},[75,76,162],[19,8512,8513,8516],{"style":648},[19,8514],{"className":8515,"style":155},[154],[19,8517,8519],{"className":8518},[159,160,161,162],[19,8520,8522,8525,8528],{"className":8521},[75,162],[19,8523,46],{"className":8524},[81,162],[19,8526,7801],{"className":8527,"style":7888},[75,76,162],[19,8529,52],{"className":8530},[88,162],[19,8532,173],{"className":8533},[172],[19,8535,8537],{"className":8536},[142],[19,8538,8540],{"className":8539,"style":4621},[146],[19,8541],{},[19,8543,46],{"className":8544},[81],[19,8546,8548,8551],{"className":8547},[75],[19,8549,49],{"className":8550},[75,76],[19,8552,8554],{"className":8553},[133],[19,8555,8557,8580],{"className":8556},[137,138],[19,8558,8560,8577],{"className":8559},[142],[19,8561,8563],{"className":8562,"style":147},[146],[19,8564,8565,8568],{"style":150},[19,8566],{"className":8567,"style":155},[154],[19,8569,8571],{"className":8570},[159,160,161,162],[19,8572,8574],{"className":8573},[75,162],[19,8575,110],{"className":8576},[75,162],[19,8578,173],{"className":8579},[172],[19,8581,8583],{"className":8582},[142],[19,8584,8586],{"className":8585,"style":180},[146],[19,8587],{},[19,8589,52],{"className":8590},[88],[19,8592,7830],{"className":8593},[8036],[19,8595,8335],{"className":8596},[411],[19,8598],{"className":8599,"style":412},[411],[19,8601,7801],{"className":8602,"style":7888},[75,76],[19,8604],{"className":8605,"style":613},[411],[19,8607,267],{"className":8608},[335],[19,8610],{"className":8611,"style":613},[411],[19,8613,8615,8619,8622,8625,8628,8631,8634,8637,8640,8643,8646,8649,8652,8655,8658],{"className":8614},[66],[19,8616],{"className":8617,"style":8618},[70],"height:0.8389em;vertical-align:-0.1944em;",[19,8620,110],{"className":8621},[75],[19,8623,7830],{"className":8624},[8036],[19,8626],{"className":8627,"style":412},[411],[19,8629,3495],{"className":8630},[75],[19,8632,7830],{"className":8633},[8036],[19,8635],{"className":8636,"style":412},[411],[19,8638,3518],{"className":8639},[75],[19,8641,7830],{"className":8642},[8036],[19,8644],{"className":8645,"style":412},[411],[19,8647,4007],{"className":8648},[75],[19,8650,7830],{"className":8651},[8036],[19,8653],{"className":8654,"style":412},[411],[19,8656,3297],{"className":8657},[75,76],[19,8659,3447],{"className":8660},[75],[15,8662,8663,8664,8894,8895,8938,8939,9007,9008,3447],{},"Returning to our conjecture, can we prove ",[19,8665,8667,8727],{"className":8666,"translate":23},[22],[19,8668,8670],{"className":8669},[27],[29,8671,8672],{"xmlns":31},[33,8673,8674,8724],{},[36,8675,8676,8678,8680,8682,8684,8686,8688,8690,8692,8694,8696,8698,8700,8702,8704,8706,8708,8710,8712,8714,8716,8722],{},[39,8677,41],{},[43,8679,46],{"stretchy":45},[39,8681,49],{},[43,8683,52],{"stretchy":45},[43,8685,267],{},[39,8687,6529],{},[43,8689,2839],{},[39,8691,3297],{},[43,8693,46],{"stretchy":45},[39,8695,49],{},[43,8697,52],{"stretchy":45},[43,8699,2351],{},[39,8701,1874],{},[43,8703,46],{"stretchy":45},[43,8705,46],{"stretchy":45},[39,8707,49],{},[43,8709,240],{},[39,8711,49],{},[43,8713,2839],{},[108,8715,110],{},[269,8717,8718,8720],{},[43,8719,52],{"stretchy":45},[39,8721,3297],{},[43,8723,52],{"stretchy":45},[54,8725,8726],{"encoding":56},"f(x)=T*{n}(x)+o((x-x*{0})^{n})",[19,8728,8730,8757,8775,8805,8829,8847],{"className":8729,"ariaHidden":62},[61],[19,8731,8733,8736,8739,8742,8745,8748,8751,8754],{"className":8732},[66],[19,8734],{"className":8735,"style":71},[70],[19,8737,41],{"className":8738,"style":77},[75,76],[19,8740,46],{"className":8741},[81],[19,8743,49],{"className":8744},[75,76],[19,8746,52],{"className":8747},[88],[19,8749],{"className":8750,"style":613},[411],[19,8752,267],{"className":8753},[335],[19,8755],{"className":8756,"style":613},[411],[19,8758,8760,8763,8766,8769,8772],{"className":8759},[66],[19,8761],{"className":8762,"style":7738},[70],[19,8764,6529],{"className":8765,"style":6724},[75,76],[19,8767],{"className":8768,"style":451},[411],[19,8770,2839],{"className":8771},[455],[19,8773],{"className":8774,"style":451},[411],[19,8776,8778,8781,8787,8790,8793,8796,8799,8802],{"className":8777},[66],[19,8779],{"className":8780,"style":71},[70],[19,8782,8784],{"className":8783},[75],[19,8785,3297],{"className":8786},[75,76],[19,8788,46],{"className":8789},[81],[19,8791,49],{"className":8792},[75,76],[19,8794,52],{"className":8795},[88],[19,8797],{"className":8798,"style":451},[411],[19,8800,2351],{"className":8801},[455],[19,8803],{"className":8804,"style":451},[411],[19,8806,8808,8811,8814,8817,8820,8823,8826],{"className":8807},[66],[19,8809],{"className":8810,"style":71},[70],[19,8812,1874],{"className":8813},[75,76],[19,8815,3368],{"className":8816},[81],[19,8818,49],{"className":8819},[75,76],[19,8821],{"className":8822,"style":451},[411],[19,8824,240],{"className":8825},[455],[19,8827],{"className":8828,"style":451},[411],[19,8830,8832,8835,8838,8841,8844],{"className":8831},[66],[19,8833],{"className":8834,"style":2854},[70],[19,8836,49],{"className":8837},[75,76],[19,8839],{"className":8840,"style":451},[411],[19,8842,2839],{"className":8843},[455],[19,8845],{"className":8846,"style":451},[411],[19,8848,8850,8853,8859,8891],{"className":8849},[66],[19,8851],{"className":8852,"style":71},[70],[19,8854,8856],{"className":8855},[75],[19,8857,110],{"className":8858},[75],[19,8860,8862,8865],{"className":8861},[88],[19,8863,52],{"className":8864},[88],[19,8866,8868],{"className":8867},[133],[19,8869,8871],{"className":8870},[137],[19,8872,8874],{"className":8873},[142],[19,8875,8877],{"className":8876,"style":3429},[146],[19,8878,8879,8882],{"style":1643},[19,8880],{"className":8881,"style":155},[154],[19,8883,8885],{"className":8884},[159,160,161,162],[19,8886,8888],{"className":8887},[75,162],[19,8889,3297],{"className":8890},[75,76,162],[19,8892,52],{"className":8893},[88],"? If this holds, then when approximating function ",[19,8896,8898,8917],{"className":8897,"translate":23},[22],[19,8899,8901],{"className":8900},[27],[29,8902,8903],{"xmlns":31},[33,8904,8905,8915],{},[36,8906,8907,8909,8911,8913],{},[39,8908,41],{},[43,8910,46],{"stretchy":45},[39,8912,49],{},[43,8914,52],{"stretchy":45},[54,8916,57],{"encoding":56},[19,8918,8920],{"className":8919,"ariaHidden":62},[61],[19,8921,8923,8926,8929,8932,8935],{"className":8922},[66],[19,8924],{"className":8925,"style":71},[70],[19,8927,41],{"className":8928,"style":77},[75,76],[19,8930,46],{"className":8931},[81],[19,8933,49],{"className":8934},[75,76],[19,8936,52],{"className":8937},[88]," with the Taylor polynomial ",[19,8940,8942,8965],{"className":8941,"translate":23},[22],[19,8943,8945],{"className":8944},[27],[29,8946,8947],{"xmlns":31},[33,8948,8949,8963],{},[36,8950,8951,8953,8955,8957,8959,8961],{},[39,8952,6529],{},[43,8954,2839],{},[39,8956,3297],{},[43,8958,46],{"stretchy":45},[39,8960,49],{},[43,8962,52],{"stretchy":45},[54,8964,7728],{"encoding":56},[19,8966,8968,8986],{"className":8967,"ariaHidden":62},[61],[19,8969,8971,8974,8977,8980,8983],{"className":8970},[66],[19,8972],{"className":8973,"style":7738},[70],[19,8975,6529],{"className":8976,"style":6724},[75,76],[19,8978],{"className":8979,"style":451},[411],[19,8981,2839],{"className":8982},[455],[19,8984],{"className":8985,"style":451},[411],[19,8987,8989,8992,8998,9001,9004],{"className":8988},[66],[19,8990],{"className":8991,"style":71},[70],[19,8993,8995],{"className":8994},[75],[19,8996,3297],{"className":8997},[75,76],[19,8999,46],{"className":9000},[81],[19,9002,49],{"className":9003},[75,76],[19,9005,52],{"className":9006},[88],", the error will be as desired, i.e., an error term higher order than ",[19,9009,9011,9041],{"className":9010,"translate":23},[22],[19,9012,9014],{"className":9013},[27],[29,9015,9016],{"xmlns":31},[33,9017,9018,9038],{},[36,9019,9020,9022,9024,9026,9028,9030,9032],{},[43,9021,46],{"stretchy":45},[39,9023,49],{},[43,9025,240],{},[39,9027,49],{},[43,9029,2839],{},[108,9031,110],{},[269,9033,9034,9036],{},[43,9035,52],{"stretchy":45},[39,9037,3297],{},[54,9039,9040],{"encoding":56},"(x-x*{0})^{n}",[19,9042,9044,9065,9083],{"className":9043,"ariaHidden":62},[61],[19,9045,9047,9050,9053,9056,9059,9062],{"className":9046},[66],[19,9048],{"className":9049,"style":71},[70],[19,9051,46],{"className":9052},[81],[19,9054,49],{"className":9055},[75,76],[19,9057],{"className":9058,"style":451},[411],[19,9060,240],{"className":9061},[455],[19,9063],{"className":9064,"style":451},[411],[19,9066,9068,9071,9074,9077,9080],{"className":9067},[66],[19,9069],{"className":9070,"style":2854},[70],[19,9072,49],{"className":9073},[75,76],[19,9075],{"className":9076,"style":451},[411],[19,9078,2839],{"className":9079},[455],[19,9081],{"className":9082,"style":451},[411],[19,9084,9086,9089,9095],{"className":9085},[66],[19,9087],{"className":9088,"style":71},[70],[19,9090,9092],{"className":9091},[75],[19,9093,110],{"className":9094},[75],[19,9096,9098,9101],{"className":9097},[88],[19,9099,52],{"className":9100},[88],[19,9102,9104],{"className":9103},[133],[19,9105,9107],{"className":9106},[137],[19,9108,9110],{"className":9109},[142],[19,9111,9113],{"className":9112,"style":3429},[146],[19,9114,9115,9118],{"style":1643},[19,9116],{"className":9117,"style":155},[154],[19,9119,9121],{"className":9120},[159,160,161,162],[19,9122,9124],{"className":9123},[75,162],[19,9125,3297],{"className":9126},[75,76,162],[15,9128,9129,9132,9133,6378,9176,6407,9204,9276],{},[7600,9130,9131],{},"Theorem:"," If the function 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Let",[19,10740,10742],{"className":10741,"translate":23},[187],[19,10743,10745,10829],{"className":10744,"translate":23},[22],[19,10746,10748],{"className":10747},[27],[29,10749,10750],{"xmlns":31,"display":196},[33,10751,10752,10826],{},[36,10753,10754,10761,10763,10765,10767,10769,10771,10773,10775,10777,10779,10785,10787,10789,10791,10793,10800,10802,10804,10806,10808,10810,10812,10814,10820],{},[103,10755,10756,10759],{},[39,10757,10758],{},"R",[39,10760,3297],{},[43,10762,46],{"stretchy":45},[39,10764,49],{},[43,10766,52],{"stretchy":45},[43,10768,267],{},[39,10770,41],{},[43,10772,46],{"stretchy":45},[39,10774,49],{},[43,10776,52],{"stretchy":45},[43,10778,240],{},[103,10780,10781,10783],{},[39,10782,6529],{},[39,10784,3297],{},[43,10786,46],{"stretchy":45},[39,10788,49],{},[43,10790,52],{"stretchy":45},[411,10792],{"linebreak":4283},[103,10794,10795,10798],{},[39,10796,10797],{},"Q",[39,10799,3297],{},[43,10801,46],{"stretchy":45},[39,10803,49],{},[43,10805,52],{"stretchy":45},[43,10807,267],{},[43,10809,46],{"stretchy":45},[39,10811,49],{},[43,10813,240],{},[103,10815,10816,10818],{},[39,10817,49],{},[108,10819,110],{},[269,10821,10822,10824],{},[43,10823,52],{"stretchy":45},[39,10825,3297],{},[54,10827,10828],{"encoding":56},"R_{n}(x)=f(x)-T_{n}(x)\\\\\nQ_{n}(x)=(x-x_{0})^{n}",[19,10830,10832,10901,10928,10986,10989,11056,11077],{"className":10831,"ariaHidden":62},[61],[19,10833,10835,10838,10883,10886,10889,10892,10895,10898],{"className":10834},[66],[19,10836],{"className":10837,"style":71},[70],[19,10839,10841,10845],{"className":10840},[75],[19,10842,10758],{"className":10843,"style":10844},[75,76],"margin-right:0.00773em;",[19,10846,10848],{"className":10847},[133],[19,10849,10851,10875],{"className":10850},[137,138],[19,10852,10854,10872],{"className":10853},[142],[19,10855,10857],{"className":10856,"style":3609},[146],[19,10858,10860,10863],{"style":10859},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[19,10861],{"className":10862,"style":155},[154],[19,10864,10866],{"className":10865},[159,160,161,162],[19,10867,10869],{"className":10868},[75,162],[19,10870,3297],{"className":10871},[75,76,162],[19,10873,173],{"className":10874},[172],[19,10876,10878],{"className":10877},[142],[19,10879,10881],{"className":10880,"style":180},[146],[19,10882],{},[19,10884,46],{"className":10885},[81],[19,10887,49],{"className":10888},[75,76],[19,10890,52],{"className":10891},[88],[19,10893],{"className":10894,"style":613},[411],[19,10896,267],{"className":10897},[335],[19,10899],{"className":10900,"style":613},[411],[19,10902,10904,10907,10910,10913,10916,10919,10922,10925],{"className":10903},[66],[19,10905],{"className":10906,"style":71},[70],[19,10908,41],{"className":10909,"style":77},[75,76],[19,10911,46],{"className":10912},[81],[19,10914,49],{"className":10915},[75,76],[19,10917,52],{"className":10918},[88],[19,10920],{"className":10921,"style":451},[411],[19,10923,240],{"className":10924},[455],[19,10926],{"className":10927,"style":451},[411],[19,10929,10931,10934,10977,10980,10983],{"className":10930},[66],[19,10932],{"className":10933,"style":71},[70],[19,10935,10937,10940],{"className":10936},[75],[19,10938,6529],{"className":10939,"style":6724},[75,76],[19,10941,10943],{"className":10942},[133],[19,10944,10946,10969],{"className":10945},[137,138],[19,10947,10949,10966],{"className":10948},[142],[19,10950,10952],{"className":10951,"style":3609},[146],[19,10953,10954,10957],{"style":6739},[19,10955],{"className":10956,"style":155},[154],[19,10958,10960],{"className":10959},[159,160,161,162],[19,10961,10963],{"className":10962},[75,162],[19,10964,3297],{"className":10965},[75,76,162],[19,10967,173],{"className":10968},[172],[19,10970,10972],{"className":10971},[142],[19,10973,10975],{"className":10974,"style":180},[146],[19,10976],{},[19,10978,46],{"className":10979},[81],[19,10981,49],{"className":10982},[75,76],[19,10984,52],{"className":10985},[88],[19,10987],{"className":10988},[411,4283],[19,10990,10992,10995,11038,11041,11044,11047,11050,11053],{"className":10991},[66],[19,10993],{"className":10994,"style":71},[70],[19,10996,10998,11001],{"className":10997},[75],[19,10999,10797],{"className":11000},[75,76],[19,11002,11004],{"className":11003},[133],[19,11005,11007,11030],{"className":11006},[137,138],[19,11008,11010,11027],{"className":11009},[142],[19,11011,11013],{"className":11012,"style":3609},[146],[19,11014,11015,11018],{"style":150},[19,11016],{"className":11017,"style":155},[154],[19,11019,11021],{"className":11020},[159,160,161,162],[19,11022,11024],{"className":11023},[75,162],[19,11025,3297],{"className":11026},[75,76,162],[19,11028,173],{"className":11029},[172],[19,11031,11033],{"className":11032},[142],[19,11034,11036],{"className":11035,"style":180},[146],[19,11037],{},[19,11039,46],{"className":11040},[81],[19,11042,49],{"className":11043},[75,76],[19,11045,52],{"className":11046},[88],[19,11048],{"className":11049,"style":613},[411],[19,11051,267],{"className":11052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Q^{(n)}_{n}(x_{0})=n!",[19,12526,12528,12635,12757,12775,12908,12917,12920,13047],{"className":12527,"ariaHidden":62},[61],[19,12529,12531,12534,12577,12580,12623,12626,12629,12632],{"className":12530},[66],[19,12532],{"className":12533,"style":71},[70],[19,12535,12537,12540],{"className":12536},[75],[19,12538,10797],{"className":12539},[75,76],[19,12541,12543],{"className":12542},[133],[19,12544,12546,12569],{"className":12545},[137,138],[19,12547,12549,12566],{"className":12548},[142],[19,12550,12552],{"className":12551,"style":3609},[146],[19,12553,12554,12557],{"style":150},[19,12555],{"className":12556,"style":155},[154],[19,12558,12560],{"className":12559},[159,160,161,162],[19,12561,12563],{"className":12562},[75,162],[19,12564,3297],{"className":12565},[75,76,162],[19,12567,173],{"className":12568},[172],[19,12570,12572],{"className":12571},[142],[19,12573,12575],{"className":12574,"style":180},[146],[19,12576],{},[19,12578,46],{"className":12579},[81],[19,12581,12583,12586],{"className":12582},[75],[19,12584,49],{"className":12585},[75,76],[19,12587,12589],{"className":12588},[133],[19,12590,12592,12615],{"className":12591},[137,138],[19,12593,12595,12612],{"className":12594},[142],[19,12596,12598],{"className":12597,"style":147},[146],[19,12599,12600,12603],{"style":150},[19,12601],{"className":12602,"style":155},[154],[19,12604,12606],{"className":12605},[159,160,161,162],[19,12607,12609],{"className":12608},[75,162],[19,12610,110],{"className":12611},[75,162],[19,12613,173],{"className":12614},[172],[19,12616,12618],{"className":12617},[142],[19,12619,12621],{"className":12620,"style":180},[146],[19,12622],{},[19,12624,52],{"className":12625},[88],[19,12627],{"className":12628,"style":613},[411],[19,12630,267],{"className":12631},[335],[19,12633],{"className":12634,"style":613},[411],[19,12636,12638,12641,12699,12702,12745,12748,12751,12754],{"className":12637},[66],[19,12639],{"className":12640,"style":626},[70],[19,12642,12644,12647],{"className":12643},[75],[19,12645,10797],{"className":12646},[75,76],[19,12648,12650],{"className":12649},[133],[19,12651,12653,12691],{"className":12652},[137,138],[19,12654,12656,12688],{"className":12655},[142],[19,12657,12659,12674],{"className":12658,"style":645},[146],[19,12660,12662,12665],{"style":12661},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[19,12663],{"className":12664,"style":155},[154],[19,12666,12668],{"className":12667},[159,160,161,162],[19,12669,12671],{"className":12670},[75,162],[19,12672,3297],{"className":12673},[75,76,162],[19,12675,12676,12679],{"style":648},[19,12677],{"className":12678,"style":155},[154],[19,12680,12682],{"className":12681},[159,160,161,162],[19,12683,12685],{"className":12684},[75,162],[19,12686,277],{"className":12687},[75,162],[19,12689,173],{"className":12690},[172],[19,12692,12694],{"className":12693},[142],[19,12695,12697],{"className":12696,"style":4621},[146],[19,12698],{},[19,12700,46],{"className":12701},[81],[19,12703,12705,12708],{"className":12704},[75],[19,12706,49],{"className":12707},[75,76],[19,12709,12711],{"className":12710},[133],[19,12712,12714,12737],{"className":12713},[137,138],[19,12715,12717,12734],{"className":12716},[142],[19,12718,12720],{"className":12719,"style":147},[146],[19,12721,12722,12725],{"style":150},[19,12723],{"className":12724,"style":155},[154],[19,12726,12728],{"className":12727},[159,160,161,162],[19,12729,12731],{"className":12730},[75,162],[19,12732,110],{"className":12733},[75,162],[19,12735,173],{"className":12736},[172],[19,12738,12740],{"className":12739},[142],[19,12741,12743],{"className":12742,"style":180},[146],[19,12744],{},[19,12746,52],{"className":12747},[88],[19,12749],{"className":12750,"style":613},[411],[19,12752,267],{"className":12753},[335],[19,12755],{"className":12756,"style":613},[411],[19,12758,12760,12763,12766,12769,12772],{"className":12759},[66],[19,12761],{"className":12762,"style":12245},[70],[19,12764,4007],{"className":12765},[75],[19,12767],{"className":12768,"style":613},[411],[19,12770,267],{"className":12771},[335],[19,12773],{"className":12774,"style":613},[411],[19,12776,12778,12781,12850,12853,12896,12899,12902,12905],{"className":12777},[66],[19,12779],{"className":12780,"style":4934},[70],[19,12782,12784,12787],{"className":12783},[75],[19,12785,10797],{"className":12786},[75,76],[19,12788,12790],{"className":12789},[133],[19,12791,12793,12842],{"className":12792},[137,138],[19,12794,12796,12839],{"className":12795},[142],[19,12797,12799,12813],{"className":12798,"style":4953},[146],[19,12800,12801,12804],{"style":12661},[19,12802],{"className":12803,"style":155},[154],[19,12805,12807],{"className":12806},[159,160,161,162],[19,12808,12810],{"className":12809},[75,162],[19,12811,3297],{"className":12812},[75,76,162],[19,12814,12815,12818],{"style":648},[19,12816],{"className":12817,"style":155},[154],[19,12819,12821],{"className":12820},[159,160,161,162],[19,12822,12824,12827,12830,12833,12836],{"className":12823},[75,162],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",[19,13063,13065,13099],{"className":13064,"translate":23},[22],[19,13066,13068],{"className":13067},[27],[29,13069,13070],{"xmlns":31},[33,13071,13072,13096],{},[36,13073,13074,13086,13088,13090,13092,13094],{},[269,13075,13076,13078],{},[39,13077,41],{},[36,13079,13080,13082,13084],{},[43,13081,46],{"stretchy":45},[39,13083,3297],{},[43,13085,52],{"stretchy":45},[43,13087,46],{"stretchy":45},[39,13089,49],{},[43,13091,2839],{},[108,13093,110],{},[43,13095,52],{"stretchy":45},[54,13097,13098],{"encoding":56},"f^{(n)}(x*{0})",[19,13100,13102,13162],{"className":13101,"ariaHidden":62},[61],[19,13103,13105,13109,13147,13150,13153,13156,13159],{"className":13104},[66],[19,13106],{"className":13107,"style":13108},[70],"height:1.138em;vertical-align:-0.25em;",[19,13110,13112,13115],{"className":13111},[75],[19,13113,41],{"className":13114,"style":77},[75,76],[19,13116,13118],{"className":13117},[133],[19,13119,13121],{"className":13120},[137],[19,13122,13124],{"className":13123},[142],[19,13125,13127],{"className":13126,"style":7416},[146],[19,13128,13129,13132],{"style":1643},[19,13130],{"className":13131,"style":155},[154],[19,13133,13135],{"className":13134},[159,160,161,162],[19,13136,13138,13141,13144],{"className":13137},[75,162],[19,13139,46],{"className":13140},[81,162],[19,13142,3297],{"className":13143},[75,76,162],[19,13145,52],{"className":13146},[88,162],[19,13148,46],{"className":13149},[81],[19,13151,49],{"className":13152},[75,76],[19,13154],{"className":13155,"style":451},[411],[19,13157,2839],{"className":13158},[455],[19,13160],{"className":13161,"style":451},[411],[19,13163,13165,13168,13174],{"className":13164},[66],[19,13166],{"className":13167,"style":71},[70],[19,13169,13171],{"className":13170},[75],[19,13172,110],{"className":13173},[75],[19,13175,52],{"className":13176},[88]," exists, ",[13179,13180],"br",{},"\nTherefore, in a neighborhood ",[19,13183,13185,13210],{"className":13184,"translate":23},[22],[19,13186,13188],{"className":13187},[27],[29,13189,13190],{"xmlns":31},[33,13191,13192,13207],{},[36,13193,13194,13197,13199,13201,13203,13205],{},[39,13195,13196],{},"U",[43,13198,46],{"stretchy":45},[39,13200,49],{},[43,13202,2839],{},[108,13204,110],{},[43,13206,52],{"stretchy":45},[54,13208,13209],{"encoding":56},"U(x*{0})",[19,13211,13213,13238],{"className":13212,"ariaHidden":62},[61],[19,13214,13216,13219,13223,13226,13229,13232,13235],{"className":13215},[66],[19,13217],{"className":13218,"style":71},[70],[19,13220,13196],{"className":13221,"style":13222},[75,76],"margin-right:0.10903em;",[19,13224,46],{"className":13225},[81],[19,13227,49],{"className":13228},[75,76],[19,13230],{"className":13231,"style":451},[411],[19,13233,2839],{"className":13234},[455],[19,13236],{"className":13237,"style":451},[411],[19,13239,13241,13244,13250],{"className":13240},[66],[19,13242],{"className":13243,"style":71},[70],[19,13245,13247],{"className":13246},[75],[19,13248,110],{"className":13249},[75],[19,13251,52],{"className":13252},[88]," of ",[19,13255,13257,13274],{"className":13256,"translate":23},[22],[19,13258,13260],{"className":13259},[27],[29,13261,13262],{"xmlns":31},[33,13263,13264,13272],{},[36,13265,13266,13268,13270],{},[39,13267,49],{},[43,13269,2839],{},[108,13271,110],{},[54,13273,2844],{"encoding":56},[19,13275,13277,13295],{"className":13276,"ariaHidden":62},[61],[19,13278,13280,13283,13286,13289,13292],{"className":13279},[66],[19,13281],{"className":13282,"style":2854},[70],[19,13284,49],{"className":13285},[75,76],[19,13287],{"className":13288,"style":451},[411],[19,13290,2839],{"className":13291},[455],[19,13293],{"className":13294,"style":451},[411],[19,13296,13298,13301],{"className":13297},[66],[19,13299],{"className":13300,"style":1214},[70],[19,13302,13304],{"className":13303},[75],[19,13305,110],{"className":13306},[75],", ",[19,13309,13311,13330],{"className":13310,"translate":23},[22],[19,13312,13314],{"className":13313},[27],[29,13315,13316],{"xmlns":31},[33,13317,13318,13328],{},[36,13319,13320,13322,13324,13326],{},[39,13321,41],{},[43,13323,46],{"stretchy":45},[39,13325,49],{},[43,13327,52],{"stretchy":45},[54,13329,57],{"encoding":56},[19,13331,13333],{"className":13332,"ariaHidden":62},[61],[19,13334,13336,13339,13342,13345,13348],{"className":13335},[66],[19,13337],{"className":13338,"style":71},[70],[19,13340,41],{"className":13341,"style":77},[75,76],[19,13343,46],{"className":13344},[81],[19,13346,49],{"className":13347},[75,76],[19,13349,52],{"className":13350},[88]," has a ",[19,13353,13355,13377],{"className":13354,"translate":23},[22],[19,13356,13358],{"className":13357},[27],[29,13359,13360],{"xmlns":31},[33,13361,13362,13374],{},[36,13363,13364,13366,13368,13370,13372],{},[43,13365,46],{"stretchy":45},[39,13367,3297],{},[43,13369,240],{},[108,13371,3495],{},[43,13373,52],{"stretchy":45},[54,13375,13376],{"encoding":56},"(n-1)",[19,13378,13380,13401],{"className":13379,"ariaHidden":62},[61],[19,13381,13383,13386,13389,13392,13395,13398],{"className":13382},[66],[19,13384],{"className":13385,"style":71},[70],[19,13387,46],{"className":13388},[81],[19,13390,3297],{"className":13391},[75,76],[19,13393],{"className":13394,"style":451},[411],[19,13396,240],{"className":13397},[455],[19,13399],{"className":13400,"style":451},[411],[19,13402,13404,13407,13410],{"className":13403},[66],[19,13405],{"className":13406,"style":71},[70],[19,13408,3495],{"className":13409},[75],[19,13411,52],{"className":13412},[88],"th order derivative ",[19,13415,13417,13451],{"className":13416,"translate":23},[22],[19,13418,13420],{"className":13419},[27],[29,13421,13422],{"xmlns":31},[33,13423,13424,13448],{},[36,13425,13426,13442,13444,13446],{},[269,13427,13428,13430],{},[39,13429,41],{},[36,13431,13432,13434,13436,13438,13440],{},[43,13433,46],{"stretchy":45},[39,13435,3297],{},[43,13437,240],{},[108,13439,3495],{},[43,13441,52],{"stretchy":45},[43,13443,46],{"stretchy":45},[39,13445,49],{},[43,13447,52],{"stretchy":45},[54,13449,13450],{"encoding":56},"f^{(n-1)}(x)",[19,13452,13454],{"className":13453,"ariaHidden":62},[61],[19,13455,13457,13460,13504,13507,13510],{"className":13456},[66],[19,13458],{"className":13459,"style":13108},[70],[19,13461,13463,13466],{"className":13462},[75],[19,13464,41],{"className":13465,"style":77},[75,76],[19,13467,13469],{"className":13468},[133],[19,13470,13472],{"className":13471},[137],[19,13473,13475],{"className":13474},[142],[19,13476,13478],{"className":13477,"style":7416},[146],[19,13479,13480,13483],{"style":1643},[19,13481],{"className":13482,"style":155},[154],[19,13484,13486],{"className":13485},[159,160,161,162],[19,13487,13489,13492,13495,13498,13501],{"className":13488},[75,162],[19,13490,46],{"className":13491},[81,162],[19,13493,3297],{"className":13494},[75,76,162],[19,13496,240],{"className":13497},[455,162],[19,13499,3495],{"className":13500},[75,162],[19,13502,52],{"className":13503},[88,162],[19,13505,46],{"className":13506},[81],[19,13508,49],{"className":13509},[75,76],[19,13511,52],{"className":13512},[88],". ",[13179,13515],{},"\nThus, when ",[19,13518,13520,13553],{"className":13519,"translate":23},[22],[19,13521,13523],{"className":13522},[27],[29,13524,13525],{"xmlns":31},[33,13526,13527,13550],{},[36,13528,13529,13531,13534,13540,13542,13544,13546,13548],{},[39,13530,49],{},[43,13532,13533],{},"∈",[269,13535,13536,13538],{},[39,13537,13196],{},[39,13539,1874],{},[43,13541,46],{"stretchy":45},[39,13543,49],{},[43,13545,2839],{},[108,13547,110],{},[43,13549,52],{"stretchy":45},[54,13551,13552],{"encoding":56},"x\\in U^{o}(x*{0})",[19,13554,13556,13575,13628],{"className":13555,"ariaHidden":62},[61],[19,13557,13559,13563,13566,13569,13572],{"className":13558},[66],[19,13560],{"className":13561,"style":13562},[70],"height:0.5782em;vertical-align:-0.0391em;",[19,13564,49],{"className":13565},[75,76],[19,13567],{"className":13568,"style":613},[411],[19,13570,13533],{"className":13571},[335],[19,13573],{"className":13574,"style":613},[411],[19,13576,13578,13581,13613,13616,13619,13622,13625],{"className":13577},[66],[19,13579],{"className":13580,"style":71},[70],[19,13582,13584,13587],{"className":13583},[75],[19,13585,13196],{"className":13586,"style":13222},[75,76],[19,13588,13590],{"className":13589},[133],[19,13591,13593],{"className":13592},[137],[19,13594,13596],{"className":13595},[142],[19,13597,13599],{"className":13598,"style":3429},[146],[19,13600,13601,13604],{"style":1643},[19,13602],{"className":13603,"style":155},[154],[19,13605,13607],{"className":13606},[159,160,161,162],[19,13608,13610],{"className":13609},[75,162],[19,13611,1874],{"className":13612},[75,76,162],[19,13614,46],{"className":13615},[81],[19,13617,49],{"className":13618},[75,76],[19,13620],{"className":13621,"style":451},[411],[19,13623,2839],{"className":13624},[455],[19,13626],{"className":13627,"style":451},[411],[19,13629,13631,13634,13640],{"className":13630},[66],[19,13632],{"className":13633,"style":71},[70],[19,13635,13637],{"className":13636},[75],[19,13638,110],{"className":13639},[75],[19,13641,52],{"className":13642},[88]," and ",[19,13645,13647,13669],{"className":13646,"translate":23},[22],[19,13648,13650],{"className":13649},[27],[29,13651,13652],{"xmlns":31},[33,13653,13654,13666],{},[36,13655,13656,13658,13660],{},[39,13657,49],{},[43,13659,218],{},[103,13661,13662,13664],{},[39,13663,49],{},[108,13665,110],{},[54,13667,13668],{"encoding":56},"x\\rightarrow x_{0}",[19,13670,13672,13690],{"className":13671,"ariaHidden":62},[61],[19,13673,13675,13678,13681,13684,13687],{"className":13674},[66],[19,13676],{"className":13677,"style":3309},[70],[19,13679,49],{"className":13680},[75,76],[19,13682],{"className":13683,"style":613},[411],[19,13685,218],{"className":13686},[335],[19,13688],{"className":13689,"style":613},[411],[19,13691,13693,13696],{"className":13692},[66],[19,13694],{"className":13695,"style":123},[70],[19,13697,13699,13702],{"className":13698},[75],[19,13700,49],{"className":13701},[75,76],[19,13703,13705],{"className":13704},[133],[19,13706,13708,13731],{"className":13707},[137,138],[19,13709,13711,13728],{"className":13710},[142],[19,13712,13714],{"className":13713,"style":147},[146],[19,13715,13716,13719],{"style":150},[19,13717],{"className":13718,"style":155},[154],[19,13720,13722],{"className":13721},[159,160,161,162],[19,13723,13725],{"className":13724},[75,162],[19,13726,110],{"className":13727},[75,162],[19,13729,173],{"className":13730},[172],[19,13732,13734],{"className":13733},[142],[19,13735,13737],{"className":13736,"style":180},[146],[19,13738],{},", by repeatedly applying L'Hôpital's rule ",[19,13741,13743,13761],{"className":13742,"translate":23},[22],[19,13744,13746],{"className":13745},[27],[29,13747,13748],{"xmlns":31},[33,13749,13750,13758],{},[36,13751,13752,13754,13756],{},[39,13753,3297],{},[43,13755,240],{},[108,13757,3495],{},[54,13759,13760],{"encoding":56},"n-1",[19,13762,13764,13782],{"className":13763,"ariaHidden":62},[61],[19,13765,13767,13770,13773,13776,13779],{"className":13766},[66],[19,13768],{"className":13769,"style":4003},[70],[19,13771,3297],{"className":13772},[75,76],[19,13774],{"className":13775,"style":451},[411],[19,13777,240],{"className":13778},[455],[19,13780],{"className":13781,"style":451},[411],[19,13783,13785,13788],{"className":13784},[66],[19,13786],{"className":13787,"style":1214},[70],[19,13789,3495],{"className":13790},[75]," times, we have",[19,13793,13795],{"className":13794,"translate":23},[187],[19,13796,13798,14295],{"className":13797,"translate":23},[22],[19,13799,13801],{"className":13800},[27],[29,13802,13803],{"xmlns":31,"display":196},[33,13804,13805,14292],{},[36,13806,13807,13827,13857,13859,13861,13881,13915,13917,13919,13921,13923,13925,13927,13929,13949,14003,14005,14007,14027,14156,14158,14160,14170,14190,14288,14290],{},[202,13808,13809,13815],{},[36,13810,13811,13813],{},[39,13812,208],{},[43,13814,211],{},[36,13816,13817,13819,13821],{},[39,13818,49],{},[43,13820,218],{},[103,13822,13823,13825],{},[39,13824,49],{},[108,13826,110],{},[226,13828,13829,13843],{},[36,13830,13831,13837,13839,13841],{},[103,13832,13833,13835],{},[39,13834,10758],{},[39,13836,3297],{},[43,13838,46],{"stretchy":45},[39,13840,49],{},[43,13842,52],{"stretchy":45},[36,13844,13845,13851,13853,13855],{},[103,13846,13847,13849],{},[39,13848,10797],{},[39,13850,3297],{},[43,13852,46],{"stretchy":45},[39,13854,49],{},[43,13856,52],{"stretchy":45},[411,13858],{"linebreak":4283},[43,13860,267],{},[202,13862,13863,13869],{},[36,13864,13865,13867],{},[39,13866,208],{},[43,13868,211],{},[36,13870,13871,13873,13875],{},[39,13872,49],{},[43,13874,218],{},[103,13876,13877,13879],{},[39,13878,49],{},[108,13880,110],{},[226,13882,13883,13899],{},[36,13884,13885,13893,13895,13897],{},[4285,13886,13887,13889,13891],{},[39,13888,10758],{},[39,13890,3297],{},[43,13892,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,13894,46],{"stretchy":45},[39,13896,49],{},[43,13898,52],{"stretchy":45},[36,13900,13901,13909,13911,13913],{},[4285,13902,13903,13905,13907],{},[39,13904,10797],{},[39,13906,3297],{},[43,13908,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,13910,46],{"stretchy":45},[39,13912,49],{},[43,13914,52],{"stretchy":45},[411,13916],{"linebreak":4283},[43,13918,267],{},[39,13920,3447],{"mathvariant":275},[39,13922,3447],{"mathvariant":275},[39,13924,3447],{"mathvariant":275},[411,13926],{"linebreak":4283},[43,13928,267],{},[202,13930,13931,13937],{},[36,13932,13933,13935],{},[39,13934,208],{},[43,13936,211],{},[36,13938,13939,13941,13943],{},[39,13940,49],{},[43,13942,218],{},[103,13944,13945,13947],{},[39,13946,49],{},[108,13948,110],{},[226,13950,13951,13977],{},[36,13952,13953,13971,13973,13975],{},[4285,13954,13955,13957,13959],{},[39,13956,10758],{},[39,13958,3297],{},[36,13960,13961,13963,13965,13967,13969],{},[43,13962,46],{"stretchy":45},[39,13964,3297],{},[43,13966,240],{},[108,13968,3495],{},[43,13970,52],{"stretchy":45},[43,13972,46],{"stretchy":45},[39,13974,49],{},[43,13976,52],{"stretchy":45},[36,13978,13979,13997,13999,14001],{},[4285,13980,13981,13983,13985],{},[39,13982,10797],{},[39,13984,3297],{},[36,13986,13987,13989,13991,13993,13995],{},[43,13988,46],{"stretchy":45},[39,13990,3297],{},[43,13992,240],{},[108,13994,3495],{},[43,13996,52],{"stretchy":45},[43,13998,46],{"stretchy":45},[39,14000,49],{},[43,14002,52],{"stretchy":45},[411,14004],{"linebreak":4283},[43,14006,267],{},[202,14008,14009,14015],{},[36,14010,14011,14013],{},[39,14012,208],{},[43,14014,211],{},[36,14016,14017,14019,14021],{},[39,14018,49],{},[43,14020,218],{},[103,14022,14023,14025],{},[39,14024,49],{},[108,14026,110],{},[226,14028,14029,14119],{},[36,14030,14031,14047,14049,14051,14053,14055,14071,14073,14079,14081,14083,14095,14097,14103,14105,14107,14109,14111,14117],{},[269,14032,14033,14035],{},[39,14034,41],{},[36,14036,14037,14039,14041,14043,14045],{},[43,14038,46],{"stretchy":45},[39,14040,3297],{},[43,14042,240],{},[108,14044,3495],{},[43,14046,52],{"stretchy":45},[43,14048,46],{"stretchy":45},[39,14050,49],{},[43,14052,52],{"stretchy":45},[43,14054,240],{},[269,14056,14057,14059],{},[39,14058,41],{},[36,14060,14061,14063,14065,14067,14069],{},[43,14062,46],{"stretchy":45},[39,14064,3297],{},[43,14066,240],{},[108,14068,3495],{},[43,14070,52],{"stretchy":45},[43,14072,46],{"stretchy":45},[103,14074,14075,14077],{},[39,14076,49],{},[108,14078,110],{},[43,14080,52],{"stretchy":45},[43,14082,240],{},[269,14084,14085,14087],{},[39,14086,41],{},[36,14088,14089,14091,14093],{},[43,14090,46],{"stretchy":45},[39,14092,3297],{},[43,14094,52],{"stretchy":45},[43,14096,46],{"stretchy":45},[103,14098,14099,14101],{},[39,14100,49],{},[108,14102,110],{},[43,14104,52],{"stretchy":45},[43,14106,46],{"stretchy":45},[39,14108,49],{},[43,14110,240],{},[103,14112,14113,14115],{},[39,14114,49],{},[108,14116,110],{},[43,14118,52],{"stretchy":45},[36,14120,14121,14123,14125,14127,14129,14131,14133,14136,14138,14140,14142,14144,14146,14148,14154],{},[39,14122,3297],{},[43,14124,46],{"stretchy":45},[39,14126,3297],{},[43,14128,240],{},[108,14130,3495],{},[43,14132,52],{"stretchy":45},[43,14134,14135],{},"⋅",[43,14137,14135],{},[43,14139,14135],{},[108,14141,3518],{},[43,14143,46],{"stretchy":45},[39,14145,49],{},[43,14147,240],{},[103,14149,14150,14152],{},[39,14151,49],{},[108,14153,110],{},[43,14155,52],{"stretchy":45},[411,14157],{"linebreak":4283},[43,14159,267],{},[226,14161,14162,14164],{},[108,14163,3495],{},[36,14165,14166,14168],{},[39,14167,3297],{},[43,14169,4341],{"stretchy":45},[202,14171,14172,14178],{},[36,14173,14174,14176],{},[39,14175,208],{},[43,14177,211],{},[36,14179,14180,14182,14184],{},[39,14181,49],{},[43,14183,218],{},[103,14185,14186,14188],{},[39,14187,49],{},[108,14189,110],{},[36,14191,14192,14195,14261,14263,14275,14277,14283,14285],{},[43,14193,14194],{"fence":62},"[",[226,14196,14197,14249],{},[36,14198,14199,14215,14217,14219,14221,14223,14239,14241,14247],{},[269,14200,14201,14203],{},[39,14202,41],{},[36,14204,14205,14207,14209,14211,14213],{},[43,14206,46],{"stretchy":45},[39,14208,3297],{},[43,14210,240],{},[108,14212,3495],{},[43,14214,52],{"stretchy":45},[43,14216,46],{"stretchy":45},[39,14218,49],{},[43,14220,52],{"stretchy":45},[43,14222,240],{},[269,14224,14225,14227],{},[39,14226,41],{},[36,14228,14229,14231,14233,14235,14237],{},[43,14230,46],{"stretchy":45},[39,14232,3297],{},[43,14234,240],{},[108,14236,3495],{},[43,14238,52],{"stretchy":45},[43,14240,46],{"stretchy":45},[103,14242,14243,14245],{},[39,14244,49],{},[108,14246,110],{},[43,14248,52],{"stretchy":45},[36,14250,14251,14253,14255],{},[39,14252,49],{},[43,14254,240],{},[103,14256,14257,14259],{},[39,14258,49],{},[108,14260,110],{},[43,14262,240],{},[269,14264,14265,14267],{},[39,14266,41],{},[36,14268,14269,14271,14273],{},[43,14270,46],{"stretchy":45},[39,14272,3297],{},[43,14274,52],{"stretchy":45},[43,14276,46],{"stretchy":45},[103,14278,14279,14281],{},[39,14280,49],{},[108,14282,110],{},[43,14284,52],{"stretchy":45},[43,14286,14287],{"fence":62},"]",[43,14289,267],{},[108,14291,110],{},[54,14293,14294],{"encoding":56},"\\lim_{x \\rightarrow x_{0}}{\\frac{R_{n}(x)}{Q_{n}(x)}}\\\\=\\lim_{x \\rightarrow x_{0}}{\\frac{R'_{n}(x)}{Q'_{n}(x)}}\\\\=...\\\\=\\lim_{x \\rightarrow x_{0}}{\\frac{R^{(n-1)}_{n}(x)}{Q^{(n-1)}_{n}(x)}}\\\\=\\lim_{x \\rightarrow x_{0}}{\\frac{f^{(n-1)}(x)-f^{(n-1)}(x_{0})-f^{(n)}(x_{0})(x-x_{0})}{n(n-1)\\cdot \\cdot \\cdot2(x-x_{0})}}\\\\=\\frac{1}{n!}\\lim_{x \\rightarrow x_{0}}{\\left[ \\frac{f^{(n-1)}(x)-f^{(n-1)}(x_{0})}{x-x_{0}}-f^{(n)}(x_{0}) 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expression proved by this theorem is termed as the ",[7600,16426,16427],{},"Taylor formula"," for function ",[19,16430,16432,16451],{"className":16431,"translate":23},[22],[19,16433,16435],{"className":16434},[27],[29,16436,16437],{"xmlns":31},[33,16438,16439,16449],{},[36,16440,16441,16443,16445,16447],{},[39,16442,41],{},[43,16444,46],{"stretchy":45},[39,16446,49],{},[43,16448,52],{"stretchy":45},[54,16450,57],{"encoding":56},[19,16452,16454],{"className":16453,"ariaHidden":62},[61],[19,16455,16457,16460,16463,16466,16469],{"className":16456},[66],[19,16458],{"className":16459,"style":71},[70],[19,16461,41],{"className":16462,"style":77},[75,76],[19,16464,46],{"className":16465},[81],[19,16467,49],{"className":16468},[75,76],[19,16470,52],{"className":16471},[88]," at ",[19,16474,16476,16493],{"className":16475,"translate":23},[22],[19,16477,16479],{"className":16478},[27],[29,16480,16481],{"xmlns":31},[33,16482,16483,16491],{},[36,16484,16485],{},[103,16486,16487,16489],{},[39,16488,49],{},[108,16490,110],{},[54,16492,113],{"encoding":56},[19,16494,16496],{"className":16495,"ariaHidden":62},[61],[19,16497,16499,16502],{"className":16498},[66],[19,16500],{"className":16501,"style":123},[70],[19,16503,16505,16508],{"className":16504},[75],[19,16506,49],{"className":16507},[75,76],[19,16509,16511],{"className":16510},[133],[19,16512,16514,16537],{"className":16513},[137,138],[19,16515,16517,16534],{"className":16516},[142],[19,16518,16520],{"className":16519,"style":147},[146],[19,16521,16522,16525],{"style":150},[19,16523],{"className":16524,"style":155},[154],[19,16526,16528],{"className":16527},[159,160,161,162],[19,16529,16531],{"className":16530},[75,162],[19,16532,110],{"className":16533},[75,162],[19,16535,173],{"className":16536},[172],[19,16538,16540],{"className":16539},[142],[19,16541,16543],{"className":16542,"style":180},[146],[19,16544],{},". Since its corresponding remainder term",[19,16547,16549],{"className":16548,"translate":23},[187],[19,16550,16552,16626],{"className":16551,"translate":23},[22],[19,16553,16555],{"className":16554},[27],[29,16556,16557],{"xmlns":31,"display":196},[33,16558,16559,16623],{},[36,16560,16561,16567,16569,16571,16573,16575,16577,16579,16581,16583,16585,16591,16593,16595,16597,16599,16601,16603,16605,16607,16609,16615,16621],{},[103,16562,16563,16565],{},[39,16564,10758],{},[39,16566,3297],{},[43,16568,46],{"stretchy":45},[39,16570,49],{},[43,16572,52],{"stretchy":45},[43,16574,267],{},[39,16576,41],{},[43,16578,46],{"stretchy":45},[39,16580,49],{},[43,16582,52],{"stretchy":45},[43,16584,240],{},[103,16586,16587,16589],{},[39,16588,6529],{},[39,16590,3297],{},[43,16592,46],{"stretchy":45},[39,16594,49],{},[43,16596,52],{"stretchy":45},[43,16598,267],{},[39,16600,1874],{},[43,16602,46],{"stretchy":45},[43,16604,46],{"stretchy":45},[39,16606,49],{},[43,16608,240],{},[103,16610,16611,16613],{},[39,16612,49],{},[108,16614,110],{},[269,16616,16617,16619],{},[43,16618,52],{"stretchy":45},[39,16620,3297],{},[43,16622,52],{"stretchy":45},[54,16624,16625],{"encoding":56},"R_{n}(x)=f(x)-T_{n}(x)=o((x-x_{0})^{n})",[19,16627,16629,16696,16723,16790,16814],{"className":16628,"ariaHidden":62},[61],[19,16630,16632,16635,16678,16681,16684,16687,16690,16693],{"className":16631},[66],[19,16633],{"className":16634,"style":71},[70],[19,16636,16638,16641],{"className":16637},[75],[19,16639,10758],{"className":16640,"style":10844},[75,76],[19,16642,16644],{"className":16643},[133],[19,16645,16647,16670],{"className":16646},[137,138],[19,16648,16650,16667],{"className":16649},[142],[19,16651,16653],{"className":16652,"style":3609},[146],[19,16654,16655,16658],{"style":10859},[19,16656],{"className":16657,"style":155},[154],[19,16659,16661],{"className":16660},[159,160,161,162],[19,16662,16664],{"className":16663},[75,162],[19,16665,3297],{"className":16666},[75,76,162],[19,16668,173],{"className":16669},[172],[19,16671,16673],{"className":16672},[142],[19,16674,16676],{"className":16675,"style":180},[146],[19,16677],{},[19,16679,46],{"className":16680},[81],[19,16682,49],{"className":16683},[75,76],[19,16685,52],{"className":16686},[88],[19,16688],{"className":16689,"style":613},[411],[19,16691,267],{"className":16692},[335],[19,16694],{"className":16695,"style":613},[411],[19,16697,16699,16702,16705,16708,16711,16714,16717,16720],{"className":16698},[66],[19,16700],{"className":16701,"style":71},[70],[19,16703,41],{"className":16704,"style":77},[75,76],[19,16706,46],{"className":16707},[81],[19,16709,49],{"className":16710},[75,76],[19,16712,52],{"className":16713},[88],[19,16715],{"className":16716,"style":451},[411],[19,16718,240],{"className":16719},[455],[19,16721],{"className":16722,"style":451},[411],[19,16724,16726,16729,16772,16775,16778,16781,16784,16787],{"className":16725},[66],[19,16727],{"className":16728,"style":71},[70],[19,16730,16732,16735],{"className":16731},[75],[19,16733,6529],{"className":16734,"style":6724},[75,76],[19,16736,16738],{"className":16737},[133],[19,16739,16741,16764],{"className":16740},[137,138],[19,16742,16744,16761],{"className":16743},[142],[19,16745,16747],{"className":16746,"style":3609},[146],[19,16748,16749,16752],{"style":6739},[19,16750],{"className":16751,"style":155},[154],[19,16753,16755],{"className":16754},[159,160,161,162],[19,16756,16758],{"className":16757},[75,162],[19,16759,3297],{"className":16760},[75,76,162],[19,16762,173],{"className":16763},[172],[19,16765,16767],{"className":16766},[142],[19,16768,16770],{"className":16769,"style":180},[146],[19,16771],{},[19,16773,46],{"className":16774},[81],[19,16776,49],{"className":16777},[75,76],[19,16779,52],{"className":16780},[88],[19,16782],{"className":16783,"style":613},[411],[19,16785,267],{"className":16786},[335],[19,16788],{"className":16789,"style":613},[411],[19,16791,16793,16796,16799,16802,16805,16808,16811],{"className":16792},[66],[19,16794],{"className":16795,"style":71},[70],[19,16797,1874],{"className":16798},[75,76],[19,16800,3368],{"className":16801},[81],[19,16803,49],{"className":16804},[75,76],[19,16806],{"className":16807,"style":451},[411],[19,16809,240],{"className":16810},[455],[19,16812],{"className":16813,"style":451},[411],[19,16815,16817,16820,16863,16895],{"className":16816},[66],[19,16818],{"className":16819,"style":71},[70],[19,16821,16823,16826],{"className":16822},[75],[19,16824,49],{"className":16825},[75,76],[19,16827,16829],{"className":16828},[133],[19,16830,16832,16855],{"className":16831},[137,138],[19,16833,16835,16852],{"className":16834},[142],[19,16836,16838],{"className":16837,"style":147},[146],[19,16839,16840,16843],{"style":150},[19,16841],{"className":16842,"style":155},[154],[19,16844,16846],{"className":16845},[159,160,161,162],[19,16847,16849],{"className":16848},[75,162],[19,16850,110],{"className":16851},[75,162],[19,16853,173],{"className":16854},[172],[19,16856,16858],{"className":16857},[142],[19,16859,16861],{"className":16860,"style":180},[146],[19,16862],{},[19,16864,16866,16869],{"className":16865},[88],[19,16867,52],{"className":16868},[88],[19,16870,16872],{"className":16871},[133],[19,16873,16875],{"className":16874},[137],[19,16876,16878],{"className":16877},[142],[19,16879,16881],{"className":16880,"style":4148},[146],[19,16882,16883,16886],{"style":648},[19,16884],{"className":16885,"style":155},[154],[19,16887,16889],{"className":16888},[159,160,161,162],[19,16890,16892],{"className":16891},[75,162],[19,16893,3297],{"className":16894},[75,76,162],[19,16896,52],{"className":16897},[88],[15,16899,16900,16901,16904,16905,3447],{},"it is also called the ",[7600,16902,16903],{},"Taylor formula with Peano remainder",". Hence, this expression is also referred to as the ",[7600,16906,16903],{},[15,16908,16909,16912,16913,16956,16957,17029],{},[7600,16910,16911],{},"Note:"," The Taylor formula (with Peano remainder) is a qualitative expression. Although this expression holds for the entire domain of ",[19,16914,16916,16935],{"className":16915,"translate":23},[22],[19,16917,16919],{"className":16918},[27],[29,16920,16921],{"xmlns":31},[33,16922,16923,16933],{},[36,16924,16925,16927,16929,16931],{},[39,16926,41],{},[43,16928,46],{"stretchy":45},[39,16930,49],{},[43,16932,52],{"stretchy":45},[54,16934,57],{"encoding":56},[19,16936,16938],{"className":16937,"ariaHidden":62},[61],[19,16939,16941,16944,16947,16950,16953],{"className":16940},[66],[19,16942],{"className":16943,"style":71},[70],[19,16945,41],{"className":16946,"style":77},[75,76],[19,16948,46],{"className":16949},[81],[19,16951,49],{"className":16952},[75,76],[19,16954,52],{"className":16955},[88],", its remainder term is meaningful only near the point ",[19,16958,16960,16977],{"className":16959,"translate":23},[22],[19,16961,16963],{"className":16962},[27],[29,16964,16965],{"xmlns":31},[33,16966,16967,16975],{},[36,16968,16969],{},[103,16970,16971,16973],{},[39,16972,49],{},[108,16974,110],{},[54,16976,113],{"encoding":56},[19,16978,16980],{"className":16979,"ariaHidden":62},[61],[19,16981,16983,16986],{"className":16982},[66],[19,16984],{"className":16985,"style":123},[70],[19,16987,16989,16992],{"className":16988},[75],[19,16990,49],{"className":16991},[75,76],[19,16993,16995],{"className":16994},[133],[19,16996,16998,17021],{"className":16997},[137,138],[19,16999,17001,17018],{"className":17000},[142],[19,17002,17004],{"className":17003,"style":147},[146],[19,17005,17006,17009],{"style":150},[19,17007],{"className":17008,"style":155},[154],[19,17010,17012],{"className":17011},[159,160,161,162],[19,17013,17015],{"className":17014},[75,162],[19,17016,110],{"className":17017},[75,162],[19,17019,173],{"className":17020},[172],[19,17022,17024],{"className":17023},[142],[19,17025,17027],{"className":17026,"style":180},[146],[19,17028],{},". Thus, this expression has strong limitations.",[10,17031,17033],{"id":17032},"taylors-theorem","Taylor's Theorem",[15,17035,17036,17037,17080,17081,17124],{},"In order to overcome the drawback of the Taylor formula above, which only allows for a qualitative analysis of functions, we need a more precise quantitative expression to characterize the function ",[19,17038,17040,17059],{"className":17039,"translate":23},[22],[19,17041,17043],{"className":17042},[27],[29,17044,17045],{"xmlns":31},[33,17046,17047,17057],{},[36,17048,17049,17051,17053,17055],{},[39,17050,41],{},[43,17052,46],{"stretchy":45},[39,17054,49],{},[43,17056,52],{"stretchy":45},[54,17058,57],{"encoding":56},[19,17060,17062],{"className":17061,"ariaHidden":62},[61],[19,17063,17065,17068,17071,17074,17077],{"className":17064},[66],[19,17066],{"className":17067,"style":71},[70],[19,17069,41],{"className":17070,"style":77},[75,76],[19,17072,46],{"className":17073},[81],[19,17075,49],{"className":17076},[75,76],[19,17078,52],{"className":17079},[88],". A quantitative expression can more accurately delineate the range of errors when approximating the function ",[19,17082,17084,17103],{"className":17083,"translate":23},[22],[19,17085,17087],{"className":17086},[27],[29,17088,17089],{"xmlns":31},[33,17090,17091,17101],{},[36,17092,17093,17095,17097,17099],{},[39,17094,41],{},[43,17096,46],{"stretchy":45},[39,17098,49],{},[43,17100,52],{"stretchy":45},[54,17102,57],{"encoding":56},[19,17104,17106],{"className":17105,"ariaHidden":62},[61],[19,17107,17109,17112,17115,17118,17121],{"className":17108},[66],[19,17110],{"className":17111,"style":71},[70],[19,17113,41],{"className":17114,"style":77},[75,76],[19,17116,46],{"className":17117},[81],[19,17119,49],{"className":17120},[75,76],[19,17122,52],{"className":17123},[88]," with a polynomial function.",[15,17126,17127,9132,17130,17173,17174,17202,17203,17256,17257,17318,17319,17371,17372,17504,17505,17581],{},[7600,17128,17129],{},"Theorem (Taylor's Theorem):",[19,17131,17133,17152],{"className":17132,"translate":23},[22],[19,17134,17136],{"className":17135},[27],[29,17137,17138],{"xmlns":31},[33,17139,17140,17150],{},[36,17141,17142,17144,17146,17148],{},[39,17143,41],{},[43,17145,46],{"stretchy":45},[39,17147,49],{},[43,17149,52],{"stretchy":45},[54,17151,57],{"encoding":56},[19,17153,17155],{"className":17154,"ariaHidden":62},[61],[19,17156,17158,17161,17164,17167,17170],{"className":17157},[66],[19,17159],{"className":17160,"style":71},[70],[19,17162,41],{"className":17163,"style":77},[75,76],[19,17165,46],{"className":17166},[81],[19,17168,49],{"className":17169},[75,76],[19,17171,52],{"className":17172},[88]," has continuous derivatives up to ",[19,17175,17177,17190],{"className":17176,"translate":23},[22],[19,17178,17180],{"className":17179},[27],[29,17181,17182],{"xmlns":31},[33,17183,17184,17188],{},[36,17185,17186],{},[39,17187,3297],{},[54,17189,3297],{"encoding":56},[19,17191,17193],{"className":17192,"ariaHidden":62},[61],[19,17194,17196,17199],{"className":17195},[66],[19,17197],{"className":17198,"style":3309},[70],[19,17200,3297],{"className":17201},[75,76],"th order on ",[19,17204,17206,17229],{"className":17205,"translate":23},[22],[19,17207,17209],{"className":17208},[27],[29,17210,17211],{"xmlns":31},[33,17212,17213,17226],{},[36,17214,17215,17217,17219,17221,17224],{},[43,17216,14194],{"stretchy":45},[39,17218,3484],{},[43,17220,7830],{"separator":62},[39,17222,17223],{},"b",[43,17225,14287],{"stretchy":45},[54,17227,17228],{"encoding":56},"[a, b]",[19,17230,17232],{"className":17231,"ariaHidden":62},[61],[19,17233,17235,17238,17241,17244,17247,17250,17253],{"className":17234},[66],[19,17236],{"className":17237,"style":71},[70],[19,17239,14194],{"className":17240},[81],[19,17242,3484],{"className":17243},[75,76],[19,17245,7830],{"className":17246},[8036],[19,17248],{"className":17249,"style":412},[411],[19,17251,17223],{"className":17252},[75,76],[19,17254,14287],{"className":17255},[88]," and has ",[19,17258,17260,17282],{"className":17259,"translate":23},[22],[19,17261,17263],{"className":17262},[27],[29,17264,17265],{"xmlns":31},[33,17266,17267,17279],{},[36,17268,17269,17271,17273,17275,17277],{},[43,17270,46],{"stretchy":45},[39,17272,3297],{},[43,17274,2351],{},[108,17276,3495],{},[43,17278,52],{"stretchy":45},[54,17280,17281],{"encoding":56},"(n+1)",[19,17283,17285,17306],{"className":17284,"ariaHidden":62},[61],[19,17286,17288,17291,17294,17297,17300,17303],{"className":17287},[66],[19,17289],{"className":17290,"style":71},[70],[19,17292,46],{"className":17293},[81],[19,17295,3297],{"className":17296},[75,76],[19,17298],{"className":17299,"style":451},[411],[19,17301,2351],{"className":17302},[455],[19,17304],{"className":17305,"style":451},[411],[19,17307,17309,17312,17315],{"className":17308},[66],[19,17310],{"className":17311,"style":71},[70],[19,17313,3495],{"className":17314},[75],[19,17316,52],{"className":17317},[88]," st order derivatives on ",[19,17320,17322,17344],{"className":17321,"translate":23},[22],[19,17323,17325],{"className":17324},[27],[29,17326,17327],{"xmlns":31},[33,17328,17329,17341],{},[36,17330,17331,17333,17335,17337,17339],{},[43,17332,46],{"stretchy":45},[39,17334,3484],{},[43,17336,7830],{"separator":62},[39,17338,17223],{},[43,17340,52],{"stretchy":45},[54,17342,17343],{"encoding":56},"(a, b)",[19,17345,17347],{"className":17346,"ariaHidden":62},[61],[19,17348,17350,17353,17356,17359,17362,17365,17368],{"className":17349},[66],[19,17351],{"className":17352,"style":71},[70],[19,17354,46],{"className":17355},[81],[19,17357,3484],{"className":17358},[75,76],[19,17360,7830],{"className":17361},[8036],[19,17363],{"className":17364,"style":412},[411],[19,17366,17223],{"className":17367},[75,76],[19,17369,52],{"className":17370},[88],", then for any given ",[19,17373,17375,17409],{"className":17374,"translate":23},[22],[19,17376,17378],{"className":17377},[27],[29,17379,17380],{"xmlns":31},[33,17381,17382,17406],{},[36,17383,17384,17386,17388,17394,17396,17398,17400,17402,17404],{},[39,17385,49],{},[43,17387,7830],{"separator":62},[103,17389,17390,17392],{},[39,17391,49],{},[108,17393,110],{},[43,17395,13533],{},[43,17397,14194],{"stretchy":45},[39,17399,3484],{},[43,17401,7830],{"separator":62},[39,17403,17223],{},[43,17405,14287],{"stretchy":45},[54,17407,17408],{"encoding":56},"x, x_{0} \\in [a, b]",[19,17410,17412,17480],{"className":17411,"ariaHidden":62},[61],[19,17413,17415,17419,17422,17425,17428,17471,17474,17477],{"className":17414},[66],[19,17416],{"className":17417,"style":17418},[70],"height:0.7335em;vertical-align:-0.1944em;",[19,17420,49],{"className":17421},[75,76],[19,17423,7830],{"className":17424},[8036],[19,17426],{"className":17427,"style":412},[411],[19,17429,17431,17434],{"className":17430},[75],[19,17432,49],{"className":17433},[75,76],[19,17435,17437],{"className":17436},[133],[19,17438,17440,17463],{"className":17439},[137,138],[19,17441,17443,17460],{"className":17442},[142],[19,17444,17446],{"className":17445,"style":147},[146],[19,17447,17448,17451],{"style":150},[19,17449],{"className":17450,"style":155},[154],[19,17452,17454],{"className":17453},[159,160,161,162],[19,17455,17457],{"className":17456},[75,162],[19,17458,110],{"className":17459},[75,162],[19,17461,173],{"className":17462},[172],[19,17464,17466],{"className":17465},[142],[19,17467,17469],{"className":17468,"style":180},[146],[19,17470],{},[19,17472],{"className":17473,"style":613},[411],[19,17475,13533],{"className":17476},[335],[19,17478],{"className":17479,"style":613},[411],[19,17481,17483,17486,17489,17492,17495,17498,17501],{"className":17482},[66],[19,17484],{"className":17485,"style":71},[70],[19,17487,14194],{"className":17488},[81],[19,17490,3484],{"className":17491},[75,76],[19,17493,7830],{"className":17494},[8036],[19,17496],{"className":17497,"style":412},[411],[19,17499,17223],{"className":17500},[75,76],[19,17502,14287],{"className":17503},[88],", there exists at least one point ",[19,17506,17508,17535],{"className":17507,"translate":23},[22],[19,17509,17511],{"className":17510},[27],[29,17512,17513],{"xmlns":31},[33,17514,17515,17532],{},[36,17516,17517,17520,17522,17524,17526,17528,17530],{},[39,17518,17519],{},"ξ",[43,17521,13533],{},[43,17523,46],{"stretchy":45},[39,17525,3484],{},[43,17527,7830],{"separator":62},[39,17529,17223],{},[43,17531,52],{"stretchy":45},[54,17533,17534],{"encoding":56},"\\xi \\in (a, b)",[19,17536,17538,17557],{"className":17537,"ariaHidden":62},[61],[19,17539,17541,17544,17548,17551,17554],{"className":17540},[66],[19,17542],{"className":17543,"style":6202},[70],[19,17545,17519],{"className":17546,"style":17547},[75,76],"margin-right:0.04601em;",[19,17549],{"className":17550,"style":613},[411],[19,17552,13533],{"className":17553},[335],[19,17555],{"className":17556,"style":613},[411],[19,17558,17560,17563,17566,17569,17572,17575,17578],{"className":17559},[66],[19,17561],{"className":17562,"style":71},[70],[19,17564,46],{"className":17565},[81],[19,17567,3484],{"className":17568},[75,76],[19,17570,7830],{"className":17571},[8036],[19,17573],{"className":17574,"style":412},[411],[19,17576,17223],{"className":17577},[75,76],[19,17579,52],{"className":17580},[88]," such that",[19,17583,17585],{"className":17584,"translate":23},[187],[19,17586,17588,17827],{"className":17587,"translate":23},[22],[19,17589,17591],{"className":17590},[27],[29,17592,17593],{"xmlns":31,"display":196},[33,17594,17595,17824],{},[36,17596,17597,17599,17601,17603,17605,17607,17609,17611,17617,17619,17621,17627,17629,17635,17637,17639,17641,17643,17649,17651,17653,17683,17685,17687,17689,17695,17701,17703,17706,17708,17740,17742,17744,17746,17752,17758,17760,17800,17802,17804,17806,17812],{},[39,17598,41],{},[43,17600,46],{"stretchy":45},[39,17602,49],{},[43,17604,52],{"stretchy":45},[43,17606,267],{},[39,17608,41],{},[43,17610,46],{"stretchy":45},[103,17612,17613,17615],{},[39,17614,49],{},[108,17616,110],{},[43,17618,52],{"stretchy":45},[43,17620,2351],{},[269,17622,17623,17625],{},[39,17624,41],{},[43,17626,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,17628,46],{"stretchy":45},[103,17630,17631,17633],{},[39,17632,49],{},[108,17634,110],{},[43,17636,52],{"stretchy":45},[43,17638,46],{"stretchy":45},[39,17640,49],{},[43,17642,240],{},[103,17644,17645,17647],{},[39,17646,49],{},[108,17648,110],{},[43,17650,52],{"stretchy":45},[43,17652,2351],{},[226,17654,17655,17677],{},[36,17656,17657,17667,17669,17675],{},[269,17658,17659,17661],{},[39,17660,41],{},[36,17662,17663,17665],{},[43,17664,277],{"mathvariant":275},[43,17666,277],{"mathvariant":275},[43,17668,46],{"stretchy":45},[103,17670,17671,17673],{},[39,17672,49],{},[108,17674,110],{},[43,17676,52],{"stretchy":45},[36,17678,17679,17681],{},[108,17680,3518],{},[43,17682,4341],{"stretchy":45},[43,17684,46],{"stretchy":45},[39,17686,49],{},[43,17688,240],{},[103,17690,17691,17693],{},[39,17692,49],{},[108,17694,110],{},[269,17696,17697,17699],{},[43,17698,52],{"stretchy":45},[108,17700,3518],{},[43,17702,2351],{},[43,17704,17705],{},"⋯",[43,17707,2351],{},[226,17709,17710,17734],{},[36,17711,17712,17724,17726,17732],{},[269,17713,17714,17716],{},[39,17715,41],{},[36,17717,17718,17720,17722],{},[43,17719,46],{"stretchy":45},[39,17721,3297],{},[43,17723,52],{"stretchy":45},[43,17725,46],{"stretchy":45},[103,17727,17728,17730],{},[39,17729,49],{},[108,17731,110],{},[43,17733,52],{"stretchy":45},[36,17735,17736,17738],{},[39,17737,3297],{},[43,17739,4341],{"stretchy":45},[43,17741,46],{"stretchy":45},[39,17743,49],{},[43,17745,240],{},[103,17747,17748,17750],{},[39,17749,49],{},[108,17751,110],{},[269,17753,17754,17756],{},[43,17755,52],{"stretchy":45},[39,17757,3297],{},[43,17759,2351],{},[226,17761,17762,17786],{},[36,17763,17764,17780,17782,17784],{},[269,17765,17766,17768],{},[39,17767,41],{},[36,17769,17770,17772,17774,17776,17778],{},[43,17771,46],{"stretchy":45},[39,17773,3297],{},[43,17775,2351],{},[108,17777,3495],{},[43,17779,52],{"stretchy":45},[43,17781,46],{"stretchy":45},[39,17783,17519],{},[43,17785,52],{"stretchy":45},[36,17787,17788,17790,17792,17794,17796,17798],{},[43,17789,46],{"stretchy":45},[39,17791,3297],{},[43,17793,2351],{},[108,17795,3495],{},[43,17797,52],{"stretchy":45},[43,17799,4341],{"stretchy":45},[43,17801,46],{"stretchy":45},[39,17803,49],{},[43,17805,240],{},[103,17807,17808,17810],{},[39,17809,49],{},[108,17811,110],{},[269,17813,17814,17816],{},[43,17815,52],{"stretchy":45},[36,17817,17818,17820,17822],{},[39,17819,3297],{},[43,17821,2351],{},[108,17823,3495],{},[54,17825,17826],{"encoding":56},"f(x)=f(x_{0})+f'(x_{0})(x-x_{0})+\\frac{f''(x_{0})}{2!}(x-x_{0})^{2}+\\dots+\\frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n}+\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-x_{0})^{n+1}",[19,17828,17830,17857,17924,18026,18087,18251,18341,18359,18529,18619,18771],{"className":17829,"ariaHidden":62},[61],[19,17831,17833,17836,17839,17842,17845,17848,17851,17854],{"className":17832},[66],[19,17834],{"className":17835,"style":71},[70],[19,17837,41],{"className":17838,"style":77},[75,76],[19,17840,46],{"className":17841},[81],[19,17843,49],{"className":17844},[75,76],[19,17846,52],{"className":17847},[88],[19,17849],{"className":17850,"style":613},[411],[19,17852,267],{"className":17853},[335],[19,17855],{"className":17856,"style":613},[411],[19,17858,17860,17863,17866,17869,17912,17915,17918,17921],{"className":17859},[66],[19,17861],{"className":17862,"style":71},[70],[19,17864,41],{"className":17865,"style":77},[75,76],[19,17867,46],{"className":17868},[81],[19,17870,17872,17875],{"className":17871},[75],[19,17873,49],{"className":17874},[75,76],[19,17876,17878],{"className":17877},[133],[19,17879,17881,17904],{"className":17880},[137,138],[19,17882,17884,17901],{"className":17883},[142],[19,17885,17887],{"className":17886,"style":147},[146],[19,17888,17889,17892],{"style":150},[19,17890],{"className":17891,"style":155},[154],[19,17893,17895],{"className":17894},[159,160,161,162],[19,17896,17898],{"className":17897},[75,162],[19,17899,110],{"className":17900},[75,162],[19,17902,173],{"className":17903},[172],[19,17905,17907],{"className":17906},[142],[19,17908,17910],{"className":17909,"style":180},[146],[19,17911],{},[19,17913,52],{"class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From the conditions of the theorem, we can see that the functions 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can be \"arbitrarily\" used on ",[19,19061,19063,19084],{"className":19062,"translate":23},[22],[19,19064,19066],{"className":19065},[27],[29,19067,19068],{"xmlns":31},[33,19069,19070,19082],{},[36,19071,19072,19074,19076,19078,19080],{},[43,19073,14194],{"stretchy":45},[39,19075,3484],{},[43,19077,7830],{"separator":62},[39,19079,17223],{},[43,19081,14287],{"stretchy":45},[54,19083,17228],{"encoding":56},[19,19085,19087],{"className":19086,"ariaHidden":62},[61],[19,19088,19090,19093,19096,19099,19102,19105,19108],{"className":19089},[66],[19,19091],{"className":19092,"style":71},[70],[19,19094,14194],{"className":19095},[81],[19,19097,3484],{"className":19098},[75,76],[19,19100,7830],{"className":19101},[8036],[19,19103],{"className":19104,"style":412},[411],[19,19106,17223],{"className":19107},[75,76],[19,19109,14287],{"className":19110},[88]," using the Mean Value Theorem, hence the proof approach utilizes the Mean Value 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both have continuous derivatives up to 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and have 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Also, 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,{"style":504},[19,24350],{"className":24351,"style":322},[154],[19,24353],{"className":24354,"style":512},[511],[19,24356,24357,24360],{"style":515},[19,24358],{"className":24359,"style":322},[154],[19,24361,24363,24407,24410,24413],{"className":24362},[75],[19,24364,24366,24369],{"className":24365},[75],[19,24367,41],{"className":24368,"style":77},[75,76],[19,24370,24372],{"className":24371},[133],[19,24373,24375],{"className":24374},[137],[19,24376,24378],{"className":24377},[142],[19,24379,24381],{"className":24380,"style":7416},[146],[19,24382,24383,24386],{"style":1643},[19,24384],{"className":24385,"style":155},[154],[19,24387,24389],{"className":24388},[159,160,161,162],[19,24390,24392,24395,24398,24401,24404],{"className":24391},[75,162],[19,24393,46],{"className":24394},[81,162],[19,24396,3297],{"className":24397},[75,76,162],[19,24399,2351],{"className":24400},[455,162],[19,24402,3495],{"className":24403},[75,162],[19,24405,52],{"className":24406},[88,162],[19,24408,46],{"className":24409},[81],[19,24411,17519],{"className":24412,"style":17547},[75,76],[19,24414,52],{"className":24415},[88],[19,24417,173],{"className":24418},[172],[19,24420,24422],{"className":24421},[142],[19,24423,24425],{"className":24424,"style":11497},[146],[19,24426],{},[19,24428],{"className":24429},[88,422],[19,24431,7830],{"className":24432},[8036],[19,24434],{"className":24435,"style":412},[411],[19,24437,17519],{"className":24438,"style":17547},[75,76],[19,24440],{"className":24441,"style":613},[411],[19,24443,13533],{"className":24444},[335],[19,24446],{"className":24447,"style":613},[411],[19,24449,24451,24454,24457,24460,24463,24466,24469],{"className":24450},[66],[19,24452],{"className":24453,"style":71},[70],[19,24455,46],{"className":24456},[81],[19,24458,3484],{"className":24459},[75,76],[19,24461,7830],{"className":24462},[8036],[19,24464],{"className":24465,"style":412},[411],[19,24467,17223],{"className":24468},[75,76],[19,24470,52],{"className":24471},[88],[15,24473,24474],{},"Proof complete.",[15,24476,24477,24479,24480,24523,24524,13253,24612,24684,24685,24713],{},[7600,24478,16911],{}," Taylor's Theorem can be alternatively stated as follows: if ",[19,24481,24483,24502],{"className":24482,"translate":23},[22],[19,24484,24486],{"className":24485},[27],[29,24487,24488],{"xmlns":31},[33,24489,24490,24500],{},[36,24491,24492,24494,24496,24498],{},[39,24493,41],{},[43,24495,46],{"stretchy":45},[39,24497,49],{},[43,24499,52],{"stretchy":45},[54,24501,57],{"encoding":56},[19,24503,24505],{"className":24504,"ariaHidden":62},[61],[19,24506,24508,24511,24514,24517,24520],{"className":24507},[66],[19,24509],{"className":24510,"style":71},[70],[19,24512,41],{"className":24513,"style":77},[75,76],[19,24515,46],{"className":24516},[81],[19,24518,49],{"className":24519},[75,76],[19,24521,52],{"className":24522},[88]," has ((n+1))st order derivatives in a neighborhood ",[19,24525,24527,24551],{"className":24526,"translate":23},[22],[19,24528,24530],{"className":24529},[27],[29,24531,24532],{"xmlns":31},[33,24533,24534,24548],{},[36,24535,24536,24538,24540,24546],{},[39,24537,13196],{},[43,24539,46],{"stretchy":45},[103,24541,24542,24544],{},[39,24543,49],{},[108,24545,110],{},[43,24547,52],{"stretchy":45},[54,24549,24550],{"encoding":56},"U(x_{0})",[19,24552,24554],{"className":24553,"ariaHidden":62},[61],[19,24555,24557,24560,24563,24566,24609],{"className":24556},[66],[19,24558],{"className":24559,"style":71},[70],[19,24561,13196],{"className":24562,"style":13222},[75,76],[19,24564,46],{"className":24565},[81],[19,24567,24569,24572],{"className":24568},[75],[19,24570,49],{"className":24571},[75,76],[19,24573,24575],{"className":24574},[133],[19,24576,24578,24601],{"className":24577},[137,138],[19,24579,24581,24598],{"className":24580},[142],[19,24582,24584],{"className":24583,"style":147},[146],[19,24585,24586,24589],{"style":150},[19,24587],{"className":24588,"style":155},[154],[19,24590,24592],{"className":24591},[159,160,161,162],[19,24593,24595],{"className":24594},[75,162],[19,24596,110],{"className":24597},[75,162],[19,24599,173],{"className":24600},[172],[19,24602,24604],{"className":24603},[142],[19,24605,24607],{"className":24606,"style":180},[146],[19,24608],{},[19,24610,52],{"className":24611},[88],[19,24613,24615,24632],{"className":24614,"translate":23},[22],[19,24616,24618],{"className":24617},[27],[29,24619,24620],{"xmlns":31},[33,24621,24622,24630],{},[36,24623,24624],{},[103,24625,24626,24628],{},[39,24627,49],{},[108,24629,110],{},[54,24631,113],{"encoding":56},[19,24633,24635],{"className":24634,"ariaHidden":62},[61],[19,24636,24638,24641],{"className":24637},[66],[19,24639],{"className":24640,"style":123},[70],[19,24642,24644,24647],{"className":24643},[75],[19,24645,49],{"className":24646},[75,76],[19,24648,24650],{"className":24649},[133],[19,24651,24653,24676],{"className":24652},[137,138],[19,24654,24656,24673],{"className":24655},[142],[19,24657,24659],{"className":24658,"style":147},[146],[19,24660,24661,24664],{"style":150},[19,24662],{"className":24663,"style":155},[154],[19,24665,24667],{"className":24666},[159,160,161,162],[19,24668,24670],{"className":24669},[75,162],[19,24671,110],{"className":24672},[75,162],[19,24674,173],{"className":24675},[172],[19,24677,24679],{"className":24678},[142],[19,24680,24682],{"className":24681,"style":180},[146],[19,24683],{},", then for any point ",[19,24686,24688,24701],{"className":24687,"translate":23},[22],[19,24689,24691],{"className":24690},[27],[29,24692,24693],{"xmlns":31},[33,24694,24695,24699],{},[36,24696,24697],{},[39,24698,49],{},[54,24700,49],{"encoding":56},[19,24702,24704],{"className":24703,"ariaHidden":62},[61],[19,24705,24707,24710],{"className":24706},[66],[19,24708],{"className":24709,"style":3309},[70],[19,24711,49],{"className":24712},[75,76]," in this neighborhood, we 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\\\\+ 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alternative description may be easier to remember.",[15,26052,26053],{},"The remainder term",[19,26055,26057],{"className":26056,"translate":23},[187],[19,26058,26060,26150],{"className":26059,"translate":23},[22],[19,26061,26063],{"className":26062},[27],[29,26064,26065],{"xmlns":31,"display":196},[33,26066,26067,26147],{},[36,26068,26069,26075,26077,26079,26081,26083,26123,26125,26127,26129,26135],{},[103,26070,26071,26073],{},[39,26072,10758],{},[39,26074,3297],{},[43,26076,46],{"stretchy":45},[39,26078,49],{},[43,26080,52],{"stretchy":45},[43,26082,267],{},[226,26084,26085,26109],{},[36,26086,26087,26103,26105,26107],{},[269,26088,26089,26091],{},[39,26090,41],{},[36,26092,26093,26095,26097,26099,26101],{},[43,26094,46],{"stretchy":45},[39,26096,3297],{},[43,26098,2351],{},[108,26100,3495],{},[43,26102,52],{"stretchy":45},[43,26104,46],{"stretchy":45},[39,26106,17519],{},[43,26108,52],{"stretchy":45},[36,26110,26111,26113,26115,26117,26119,26121],{},[43,26112,46],{"stretchy":45},[39,26114,3297],{},[43,26116,2351],{},[108,26118,3495],{},[43,26120,52],{"stretchy":45},[43,26122,4341],{"stretchy":45},[43,26124,46],{"stretchy":45},[39,26126,49],{},[43,26128,240],{},[103,26130,26131,26133],{},[39,26132,49],{},[108,26134,110],{},[269,26136,26137,26139],{},[43,26138,52],{"stretchy":45},[36,26140,26141,26143,26145],{},[39,26142,3297],{},[43,26144,2351],{},[108,26146,3495],{},[54,26148,26149],{"encoding":56},"R_{n}(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-x_{0})^{n+1}",[19,26151,26153,26220,26371],{"className":26152,"ariaHidden":62},[61],[19,26154,26156,26159,26202,26205,26208,26211,26214,26217],{"className":26155},[66],[19,26157],{"className":26158,"style":71},[70],[19,26160,26162,26165],{"className":26161},[75],[19,26163,10758],{"className":26164,"style":10844},[75,76],[19,26166,26168],{"className":26167},[133],[19,26169,26171,26194],{"className":26170},[137,138],[19,26172,26174,26191],{"className":26173},[142],[19,26175,26177],{"className":26176,"style":3609},[146],[19,26178,26179,26182],{"style":10859},[19,26180],{"className":26181,"style":155},[154],[19,26183,26185],{"className":26184},[159,160,161,162],[19,26186,26188],{"className":26187},[75,162],[19,26189,3297],{"className":26190},[75,76,162],[19,26192,173],{"className":26193},[172],[19,26195,26197],{"className":26196},[142],[19,26198,26200],{"className":26199,"style":180},[146],[19,26201],{},[19,26203,46],{"className":26204},[81],[19,26206,49],{"className":26207},[75,76],[19,26209,52],{"className":26210},[88],[19,26212],{"className":26213,"style":613},[411],[19,26215,267],{"className":26216},[335],[19,26218],{"className":26219,"style":613},[411],[19,26221,26223,26226,26356,26359,26362,26365,26368],{"className":26222},[66],[19,26224],{"className":26225,"style":15256},[70],[19,26227,26229,26232,26353],{"className":26228},[75],[19,26230],{"className":26231},[81,422],[19,26233,26235],{"className":26234},[226],[19,26236,26238,26345],{"className":26237},[137,138],[19,26239,26241,26342],{"className":26240},[142],[19,26242,26244,26273,26281],{"className":26243,"style":7367},[146],[19,26245,26246,26249],{"style":438},[19,26247],{"className":26248,"style":322},[154],[19,26250,26252,26255,26258,26261,26264,26267,26270],{"className":26251},[75],[19,26253,46],{"className":26254},[81],[19,26256,3297],{"className":26257},[75,76],[19,26259],{"className":26260,"style":451},[411],[19,26262,2351],{"className":26263},[455],[19,26265],{"className":26266,"style":451},[411],[19,26268,3495],{"className":26269},[75],[19,26271,18672],{"className":26272},[88],[19,26274,26275,26278],{"style":504},[19,26276],{"className":26277,"style":322},[154],[19,26279],{"className":26280,"style":512},[511],[19,26282,26283,26286],{"style":515},[19,26284],{"className":26285,"style":322},[154],[19,26287,26289,26333,26336,26339],{"className":26288},[75],[19,26290,26292,26295],{"className":26291},[75],[19,26293,41],{"className":26294,"style":77},[75,76],[19,26296,26298],{"className":26297},[133],[19,26299,26301],{"className":26300},[137],[19,26302,26304],{"className":26303},[142],[19,26305,26307],{"className":26306,"style":7416},[146],[19,26308,26309,26312],{"style":1643},[19,26310],{"className":26311,"style":155},[154],[19,26313,26315],{"className":26314},[159,160,161,162],[19,26316,26318,26321,26324,26327,26330],{"className":26317},[75,162],[19,26319,46],{"className":26320},[81,162],[19,26322,3297],{"className":26323},[75,76,162],[19,26325,2351],{"className":26326},[455,162],[19,26328,3495],{"className":26329},[75,162],[19,26331,52],{"className":26332},[88,162],[19,26334,46],{"className":26335},[81],[19,26337,17519],{"className":26338,"style":17547},[75,76],[19,26340,52],{"className":26341},[88],[19,26343,173],{"className":26344},[172],[19,26346,26348],{"className":26347},[142],[19,26349,26351],{"className":26350,"style":11497},[146],[19,26352],{},[19,26354],{"className":26355},[88,422],[19,26357,46],{"className":26358},[81],[19,26360,49],{"className":26361},[75,76],[19,26363],{"className":26364,"style":451},[411],[19,26366,240],{"className":26367},[455],[19,26369],{"className":26370,"st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called the ",[7600,26462,26463],{},"Lagrange remainder",", hence Taylor's Theorem can also be called the ",[7600,26466,26467],{},"Taylor formula with Lagrange remainder",[15,26469,26470,26471,26499,26500,26572,26573,26616],{},"Furthermore, from the conditions of Taylor's Theorem, it can be seen that the conditions for its use are much stricter compared to the Taylor formula (which only requires ",[19,26472,26474,26487],{"className":26473,"translate":23},[22],[19,26475,26477],{"className":26476},[27],[29,26478,26479],{"xmlns":31},[33,26480,26481,26485],{},[36,26482,26483],{},[39,26484,3297],{},[54,26486,3297],{"encoding":56},[19,26488,26490],{"className":26489,"ariaHidden":62},[61],[19,26491,26493,26496],{"className":26492},[66],[19,26494],{"className":26495,"style":3309},[70],[19,26497,3297],{"className":26498},[75,76],"th order derivatives to exist at a point ",[19,26501,26503,26520],{"className":26502,"translate":23},[22],[19,26504,26506],{"className":26505},[27],[29,26507,26508],{"xmlns":31},[33,26509,26510,26518],{},[36,26511,26512],{},[103,26513,26514,26516],{},[39,26515,49],{},[108,26517,110],{},[54,26519,113],{"encoding":56},[19,26521,26523],{"className":26522,"ariaHidden":62},[61],[19,26524,26526,26529],{"className":26525},[66],[19,26527],{"className":26528,"style":123},[70],[19,26530,26532,26535],{"className":26531},[75],[19,26533,49],{"className":26534},[75,76],[19,26536,26538],{"className":26537},[133],[19,26539,26541,26564],{"className":26540},[137,138],[19,26542,26544,26561],{"className":26543},[142],[19,26545,26547],{"className":26546,"style":147},[146],[19,26548,26549,26552],{"style":150},[19,26550],{"className":26551,"style":155},[154],[19,26553,26555],{"className":26554},[159,160,161,162],[19,26556,26558],{"className":26557},[75,162],[19,26559,110],{"className":26560},[75,162],[19,26562,173],{"className":26563},[172],[19,26565,26567],{"className":26566},[142],[19,26568,26570],{"className":26569,"style":180},[146],[19,26571],{},"), hence the conclusion obtained is stronger. Taylor's Theorem can be utilized to quantitatively approximate the function ",[19,26574,26576,26595],{"className":26575,"translate":23},[22],[19,26577,26579],{"className":26578},[27],[29,26580,26581],{"xmlns":31},[33,26582,26583,26593],{},[36,26584,26585,26587,26589,26591],{},[39,26586,41],{},[43,26588,46],{"stretchy":45},[39,26590,49],{},[43,26592,52],{"stretchy":45},[54,26594,57],{"encoding":56},[19,26596,26598],{"className":26597,"ariaHidden":62},[61],[19,26599,26601,26604,26607,26610,26613],{"className":26600},[66],[19,26602],{"className":26603,"style":71},[70],[19,26605,41],{"className":26606,"style":77},[75,76],[19,26608,46],{"className":26609},[81],[19,26611,49],{"className":26612},[75,76],[19,26614,52],{"className":26615},[88]," using polynomial functions.",[10,26618,26620],{"id":26619},"taylor-series","Taylor Series",[15,26622,26623],{},"The Taylor series, compared to the two mathematical concepts mentioned earlier, appeared later in mathematical analysis because it requires a foundation in knowledge of series, power series, and function series. Since the knowledge of series and function series is not the focus of this article, interested readers can refer to any textbook on mathematical analysis for further study. Here, we only provide information closely related to the Taylor series, focusing on power series.",[15,26625,26626,26627,26823],{},"A power series is a type of function series with the simplest form, generated by the sequence of functions ",[19,26628,26630,26672],{"className":26629,"translate":23},[22],[19,26631,26633],{"className":26632},[27],[29,26634,26635],{"xmlns":31},[33,26636,26637,26669],{},[36,26638,26639,26642,26648,26650,26652,26654,26660,26666],{},[43,26640,26641],{"fence":62},"{",[103,26643,26644,26646],{},[39,26645,3484],{},[39,26647,3297],{},[43,26649,46],{"stretchy":45},[39,26651,49],{},[43,26653,240],{},[103,26655,26656,26658],{},[39,26657,49],{},[108,26659,110],{},[269,26661,26662,26664],{},[43,26663,52],{"stretchy":45},[39,26665,3297],{},[43,26667,26668],{"fence":62},"}",[54,26670,26671],{"encoding":56},"\\left\\{ a_{n}(x-x_{0})^{n} \\right\\}",[19,26673,26675],{"className":26674,"ariaHidden":62},[61],[19,26676,26678,26681],{"className":26677},[66],[19,26679],{"className":26680,"style":71},[70],[19,26682,26684,26687,26730,26733,26736,26739,26742,26745,26788,26820],{"className":26683},[16026],[19,26685,26641],{"className":26686,"style":16031},[81,16030],[19,26688,26690,26693],{"className":26689},[75],[19,26691,3484],{"className":26692},[75,76],[19,26694,26696],{"className":26695},[133],[19,26697,26699,26722],{"className":26698},[137,138],[19,26700,26702,26719],{"className":26701},[142],[19,26703,26705],{"className":26704,"style":3609},[146],[19,26706,26707,26710],{"style":150},[19,26708],{"className":26709,"style":155},[154],[19,26711,26713],{"className":26712},[159,160,161,162],[19,26714,26716],{"className":26715},[75,162],[19,26717,3297],{"className":26718},[75,76,162],[19,26720,173],{"className":26721},[172],[19,26723,26725],{"className":26724},[142],[19,26726,26728],{"className":26727,"style":180},[146],[19,26729],{},[19,26731,46],{"className":26732},[81],[19,26734,49],{"className":26735},[75,76],[19,26737],{"className":26738,"style":451},[411],[19,26740,240],{"className":26741},[455],[19,26743],{"className":26744,"style":451},[411],[19,26746,26748,26751],{"className":26747},[75],[19,26749,49],{"className":26750},[75,76],[19,26752,26754],{"className":26753},[133],[19,26755,26757,26780],{"className":26756},[137,138],[19,26758,26760,26777],{"className":26759},[142],[19,26761,26763],{"className":26762,"style":147},[146],[19,26764,26765,26768],{"style":150},[19,26766],{"className":26767,"style":155},[154],[19,26769,26771],{"className":26770},[159,160,161,162],[19,26772,26774],{"className":26773},[75,162],[19,26775,110],{"className":26776},[75,162],[19,26778,173],{"className":26779},[172],[19,26781,26783],{"className":26782},[142],[19,26784,26786],{"className":26785,"style":180},[146],[19,26787],{},[19,26789,26791,26794],{"className":26790},[88],[19,26792,52],{"className":26793},[88],[19,26795,26797],{"className":26796},[133],[19,26798,26800],{"className":26799},[137],[19,26801,26803],{"className":26802},[142],[19,26804,26806],{"className":26805,"style":3429},[146],[19,26807,26808,26811],{"style":1643},[19,26809],{"className":26810,"style":155},[154],[19,26812,26814],{"className":26813},[159,160,161,162],[19,26815,26817],{"className":26816},[75,162],[19,26818,3297],{"className":26819},[75,76,162],[19,26821,26668],{"className":26822,"style":16031},[88,16030],":",[19,26825,26827],{"className":26826,"translate":23},[187],[19,26828,26830,26975],{"className":26829,"translate":23},[22],[19,26831,26833],{"className":26832},[27],[29,26834,26835],{"xmlns":31,"display":196},[33,26836,26837,26972],{},[36,26838,26839,26856,26882,26884,26890,26892,26898,26900,26902,26904,26910,26912,26914,26920,26922,26924,26926,26932,26938,26940,26942,26944,26950,26952,26954,26956,26962,26968,26970],{},[26840,26841,26842,26845,26853],"munderover",{},[43,26843,26844],{},"∑",[36,26846,26847,26849,26851],{},[39,26848,3297],{},[43,26850,267],{},[108,26852,110],{},[39,26854,26855],{"mathvariant":275},"∞",[36,26857,26858,26864,26866,26868,26870,26876],{},[103,26859,26860,26862],{},[39,26861,3484],{},[39,26863,3297],{},[43,26865,46],{"stretchy":45},[39,26867,49],{},[43,26869,240],{},[103,26871,26872,26874],{},[39,26873,49],{},[108,26875,110],{},[269,26877,26878,26880],{},[43,26879,52],{"stretchy":45},[39,26881,3297],{},[43,26883,267],{},[103,26885,26886,26888],{},[39,26887,3484],{},[108,26889,110],{},[43,26891,2351],{},[103,26893,26894,26896],{},[39,26895,3484],{},[108,26897,3495],{},[43,26899,46],{"stretchy":45},[39,26901,49],{},[43,26903,240],{},[103,26905,26906,26908],{},[39,26907,49],{},[108,26909,110],{},[43,26911,52],{"stretchy":45},[43,26913,2351],{},[103,26915,26916,26918],{},[39,26917,3484],{},[108,26919,3518],{},[43,26921,46],{"stretchy":45},[39,26923,49],{},[43,26925,240],{},[103,26927,26928,26930],{},[39,26929,49],{},[108,26931,110],{},[269,26933,26934,26936],{},[43,26935,52],{"stretchy":45},[108,26937,3518],{},[43,26939,2351],{},[43,26941,17705],{},[43,26943,2351],{},[103,26945,26946,26948],{},[39,26947,3484],{},[39,26949,3297],{},[43,26951,46],{"stretchy":45},[39,26953,49],{},[43,26955,240],{},[103,26957,26958,26960],{},[39,26959,49],{},[108,26961,110],{},[269,26963,26964,26966],{},[43,26965,52],{"stretchy":45},[39,26967,3297],{},[43,26969,2351],{},[43,26971,21943],{},[54,26973,26974],{"encoding":56},"\\sum_{n=0}^{\\infty}{a_{n}(x-x_{0})^{n}}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\\dots+a_{n}(x-x_{0})^{n}+\\dots",[19,26976,26978,27208,27266,27330,27391,27455,27545,27563,27627,27717],{"className":26977,"ariaHidden":62},[61],[19,26979,26981,26985,27060,27063,27199,27202,27205],{"className":26980},[66],[19,26982],{"className":26983,"style":26984},[70],"height:2.9185em;vertical-align:-1.2671em;",[19,26986,26988],{"className":26987},[304,305],[19,26989,26991,27051],{"className":26990},[137,138],[19,26992,26994,27048],{"className":26993},[142],[19,26995,26998,270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simplify the form, we only discuss the power series when ",[19,27730,27732,27754],{"className":27731,"translate":23},[22],[19,27733,27735],{"className":27734},[27],[29,27736,27737],{"xmlns":31},[33,27738,27739,27751],{},[36,27740,27741,27747,27749],{},[103,27742,27743,27745],{},[39,27744,49],{},[108,27746,110],{},[43,27748,267],{},[108,27750,110],{},[54,27752,27753],{"encoding":56},"x_{0}=0",[19,27755,27757,27815],{"className":27756,"ariaHidden":62},[61],[19,27758,27760,27763,27806,27809,27812],{"className":27759},[66],[19,27761],{"className":27762,"style":123},[70],[19,27764,27766,27769],{"className":27765},[75],[19,27767,49],{"className":27768},[75,76],[19,27770,27772],{"className":27771},[133],[19,27773,27775,27798],{"className":27774},[137,138],[19,27776,27778,27795],{"className":27777},[142],[19,27779,27781],{"className":27780,"style":147},[146],[19,27782,27783,27786],{"style":150},[19,27784],{"className":27785,"style":155},[154],[19,27787,27789],{"className":27788},[159,160,161,162],[19,27790,27792],{"className":27791},[75,162],[19,27793,110],{"className":27794},[75,162],[19,27796,173],{"className":27797},[172],[19,27799,27801],{"className":27800},[142],[19,27802,27804],{"className":27803,"style":180},[146],[19,27805],{},[19,27807],{"className":27808,"style":613},[411],[19,27810,267],{"className":27811},[335],[19,27813],{"className":27814,"style":613},[411],[19,27816,27818,27821],{"className":27817},[66],[19,27819],{"className":27820,"style":1214},[70],[19,27822,110],{"className":27823},[75],[19,27825,27827],{"className":27826,"translate":23},[187],[19,27828,27830,27924],{"className":27829,"translate":23},[22],[19,27831,27833],{"className":27832},[27],[29,27834,27835],{"xmlns":31,"display":196},[33,27836,27837,27921],{},[36,27838,27839,27853,27867,27869,27875,27877,27883,27885,27887,27893,27899,27901,27903,27905,27911,27917,27919],{},[26840,27840,27841,27843,27851],{},[43,27842,26844],{},[36,27844,27845,27847,27849],{},[39,27846,3297],{},[43,27848,267],{},[108,27850,110],{},[39,27852,26855],{"mathvariant":275},[36,27854,27855,27861],{},[103,27856,27857,27859],{},[39,27858,3484],{},[39,27860,3297],{},[269,27862,27863,27865],{},[39,27864,49],{},[39,27866,3297],{},[43,27868,267],{},[103,27870,27871,27873],{},[39,27872,3484],{},[108,27874,110],{},[43,27876,2351],{},[103,27878,27879,27881],{},[39,27880,3484],{},[108,27882,3495],{},[39,27884,49],{},[43,27886,2351],{},[103,27888,27889,27891],{},[39,27890,3484],{},[108,27892,3518],{},[269,27894,27895,27897],{},[39,27896,49],{},[108,27898,3518],{},[43,27900,2351],{},[43,27902,17705],{},[43,27904,2351],{},[103,27906,27907,27909],{},[39,27908,3484],{},[39,27910,3297],{},[269,27912,27913,27915],{},[39,27914,49],{},[39,27916,3297],{},[43,27918,2351],{},[43,27920,21943],{},[54,27922,27923],{"encoding":56},"\\sum_{n=0}^{\\infty}{a_{n}x^{n}}=a_{0}+a_{1}x+a_{2}x^{2}+\\dots+a_{n}x^{n}+\\dots",[19,27925,27927,28090,28148,28209,28300,28318,28409],{"className":27926,"ariaHidden":62},[61],[19,27928,27930,27933,28000,28003,28081,28084,28087],{"className":27929},[66],[19,27931],{"className":27932,"style":26984},[70],[19,27934,27936],{"className":27935},[304,305],[19,27937,27939,27992],{"className":27938},[137,138],[19,27940,27942,27989],{"className":27941},[142],[19,27943,27945,27965,27975],{"className":27944,"style":26997},[146],[19,27946,27947,27950],{"style":27000},[19,27948],{"className":27949,"style":27004},[154],[19,27951,27953],{"className":27952},[159,160,161,162],[19,27954,27956,27959,27962],{"className":27955},[75,162],[19,27957,3297],{"className":27958},[75,76,162],[19,27960,267],{"className":27961},[335,162],[19,27963,110],{"className":27964},[75,162],[19,27966,27967,27970],{"style":27022},[19,27968],{"className":27969,"style":27004},[154],[19,27971,27972],{},[19,27973,26844],{"className":27974},[304,27031,27032],[19,27976,27977,27980],{"style":27035},[19,27978],{"className":27979,"style":27004},[154],[19,27981,27983],{"className":27982},[159,160,161,162],[19,27984,27986],{"className":27985},[75,162],[19,27987,26855],{"className":27988},[75,162],[19,27990,173],{"className":27991},[172],[19,27993,27995],{"className":27994},[142],[19,27996,27998],{"className":27997,"style":27057},[146],[19,27999],{},[19,28001],{"className":28002,"style":412},[411],[19,28004,28006,28049],{"className":28005},[75],[19,28007,28009,28012],{"className":28008},[75],[19,28010,3484],{"className":28011},[75,76],[19,28013,28015],{"className":28014},[133],[19,28016,28018,28041],{"className":28017},[137,138],[19,28019,28021,28038],{"className":28020},[142],[19,28022,28024],{"className":28023,"style":3609},[146],[19,28025,28026,28029],{"style":150},[19,28027],{"className":28028,"style":155},[154],[19,28030,28032],{"className":28031},[159,160,161,162],[19,28033,28035],{"className":28034},[75,162],[19,28036,3297],{"className":28037},[75,76,162],[19,28039,173],{"className":28040},[172],[19,28042,28044],{"className":28043},[142],[19,28045,28047],{"className":28046,"style":180},[146],[19,28048],{},[19,28050,28052,28055],{"className":28051},[75],[19,28053,49],{"className":28054},[75,76],[19,28056,28058],{"className":28057},[133],[19,28059,28061],{"className":28060},[137],[19,28062,28064],{"className":28063},[142],[19,28065,28067],{"className":28066,"style":4148},[146],[19,28068,28069,28072],{"style":648},[19,28070],{"className":28071,"style":155},[154],[19,28073,28075],{"className":28074},[159,160,161,162],[19,28076,28078],{"className":28077},[75,162],[19,28079,3297],{"className":28080},[75,76,162],[19,28082],{"className":28083,"style":613},[411],[19,28085,267],{"className":28086},[335],[19,28088],{"className":28089,"style":613},[411],[19,28091,28093,28096,28139,28142,28145],{"className":28092},[66],[19,28094],{"className":28095,"style":3660},[70],[19,28097,28099,28102],{"className":28098},[75],[19,28100,3484],{"className":28101},[75,76],[19,28103,28105],{"className":28104},[133],[19,28106,28108,28131],{"className":28107},[137,138],[19,28109,28111,28128],{"className":28110},[142],[19,28112,28114],{"className":28113,"style":147},[146],[19,28115,28116,28119],{"style":150},[19,28117],{"className":28118,"style":155},[154],[19,28120,28122],{"className":28121},[159,160,161,162],[19,28123,28125],{"className":28124},[75,162],[19,28126,110],{"className":28127},[75,162],[19,28129,173],{"className":28130},[172],[19,28132,28134],{"className":28133},[142],[19,28135,28137],{"className":28136,"style":180},[146],[19,28138],{},[19,28140],{"className":28141,"style":451},[411],[19,28143,2351],{"className":28144},[455],[19,28146],{"className":28147,"style":451},[411],[19,28149,28151,28154,28197,28200,28203,28206],{"className":28150},[66],[19,28152],{"className":28153,"style":3660},[70],[19,28155,28157,28160],{"className":28156},[75],[19,28158,3484],{"className":28159},[75,76],[19,28161,28163],{"className":28162},[133],[19,28164,28166,28189],{"className":28165},[137,138],[19,28167,28169,28186],{"className":28168},[142],[19,28170,28172],{"className":28171,"style":147},[146],[19,28173,28174,28177],{"style":150},[19,28175],{"className":28176,"style":155},[154],[19,28178,28180],{"className":28179},[159,160,161,162],[19,28181,28183],{"className":28182},[75,162],[19,28184,3495],{"className":28185},[75,162],[19,28187,173],{"className":28188},[172],[19,28190,28192],{"className":28191},[142],[19,28193,28195],{"className":28194,"style":180},[146],[19,28196],{},[19,28198,49],{"className":28199},[75,76],[19,28201],{"className":28202,"style":451},[411],[19,28204,2351],{"className":28205},[455],[19,28207],{"className":28208,"style":451},[411],[19,28210,28212,28216,28259,28291,28294,28297],{"className":28211},[66],[19,28213],{"className":28214,"style":28215},[70],"height:1.0141em;vertical-align:-0.15em;",[19,28217,28219,28222],{"className":28218},[75],[19,28220,3484],{"className":28221},[75,76],[19,28223,28225],{"className":28224},[133],[19,28226,28228,28251],{"className":28227},[137,138],[19,28229,28231,28248],{"className":28230},[142],[19,28232,28234],{"className":28233,"style":147},[146],[19,28235,28236,28239],{"style":150},[19,28237],{"className":28238,"style":155},[154],[19,28240,28242],{"className":28241},[159,160,161,162],[19,28243,28245],{"className":28244},[75,162],[19,28246,3518],{"className":28247},[75,162],[19,28249,173],{"className":28250},[172],[19,28252,28254],{"className":28253},[142],[19,28255,28257],{"className":28256,"style":180},[146],[19,28258],{},[19,28260,28262,28265],{"className":28261},[75],[19,28263,49],{"className":28264},[75,76],[19,28266,28268],{"className":28267},[133],[19,28269,28271],{"className":28270},[137],[19,28272,28274],{"className":28273},[142],[19,28275,28277],{"className":28276,"style":3970},[146],[19,28278,28279,28282],{"style":648},[19,28280],{"className":28281,"style":155},[154],[19,28283,28285],{"className":28284},[159,160,161,162],[19,28286,28288],{"className":28287},[75,162],[19,28289,3518],{"className":28290},[75,162],[19,28292],{"className":28293,"style":451},[411],[19,28295,2351],{"className":28296},[455],[19,28298],{"className":28299,"style":451},[411],[19,28301,28303,28306,28309,28312,28315],{"className":28302},[66],[19,28304],{"className":28305,"style":4003},[70],[19,28307,17705],{"className":28308},[16026],[19,28310],{"className":28311,"style":451},[411],[19,28313,2351],{"className":28314},[455],[19,28316],{"className":28317,"style":451},[411],[19,28319,28321,28325,28368,28400,28403,28406],{"className":28320},[66],[19,28322],{"className":28323,"style":28324},[70],"height:0.8644em;vertical-align:-0.15em;",[19,28326,28328,28331],{"className":28327},[75],[19,28329,3484],{"className":28330},[75,76],[19,28332,28334],{"className":28333},[133],[19,28335,28337,28360],{"className":28336},[137,138],[19,28338,28340,28357],{"className":28339},[142],[19,28341,28343],{"className":28342,"style":3609},[146],[19,28344,28345,28348],{"style":150},[19,28346],{"className":28347,"style":155},[154],[19,28349,28351],{"className":28350},[159,160,161,162],[19,28352,28354],{"className":28353},[75,162],[19,28355,3297],{"className":28356},[75,76,162],[19,28358,173],{"className":28359},[172],[19,28361,28363],{"className":28362},[142],[19,28364,28366],{"className":28365,"style":180},[146],[19,28367],{},[19,28369,28371,28374],{"className":28370},[75],[19,28372,49],{"className":28373},[75,76],[19,28375,28377],{"className":28376},[133],[19,28378,28380],{"className":28379},[137],[19,28381,28383],{"className":28382},[142],[19,28384,28386],{"className":28385,"style":4148},[146],[19,28387,28388,28391],{"style":648},[19,28389],{"className":28390,"style":155},[154],[19,28392,28394],{"className":28393},[159,160,161,162],[19,28395,28397],{"className":28396},[75,162],[19,28398,3297],{"className":28399},[75,76,162],[19,28401],{"className":28402,"style":451},[411],[19,28404,2351],{"className":28405},[455],[19,28407],{"className":28408,"style":451},[411],[19,28410,28412,28415],{"className":28411},[66],[19,28413],{"className":28414,"style":23412},[70],[19,28416,21943],{"className":28417},[16026],[15,28419,28420,28421,28449,28450,28545],{},"Correspondingly, by replacing ",[19,28422,28424,28437],{"className":28423,"translate":23},[22],[19,28425,28427],{"className":28426},[27],[29,28428,28429],{"xmlns":31},[33,28430,28431,28435],{},[36,28432,28433],{},[39,28434,49],{},[54,28436,49],{"encoding":56},[19,28438,28440],{"className":28439,"ariaHidden":62},[61],[19,28441,28443,28446],{"className":28442},[66],[19,28444],{"className":28445,"style":3309},[70],[19,28447,49],{"className":28448},[75,76]," with ",[19,28451,28453,28475],{"className":28452,"translate":23},[22],[19,28454,28456],{"className":28455},[27],[29,28457,28458],{"xmlns":31},[33,28459,28460,28472],{},[36,28461,28462,28464,28466],{},[39,28463,49],{},[43,28465,240],{},[103,28467,28468,28470],{},[39,28469,49],{},[108,28471,110],{},[54,28473,28474],{"encoding":56},"x-x_{0}",[19,28476,28478,28496],{"className":28477,"ariaHidden":62},[61],[19,28479,28481,28484,28487,28490,28493],{"className":28480},[66],[19,28482],{"className":28483,"style":4003},[70],[19,28485,49],{"className":28486},[75,76],[19,28488],{"className":28489,"style":451},[411],[19,28491,240],{"className":28492},[455],[19,28494],{"className":28495,"style":451},[411],[19,28497,28499,28502],{"className":28498},[66],[19,28500],{"className":28501,"style":123},[70],[19,28503,28505,28508],{"className":28504},[75],[19,28506,49],{"className":28507},[75,76],[19,28509,28511],{"className":28510},[133],[19,28512,28514,28537],{"className":28513},[137,138],[19,28515,28517,28534],{"className":28516},[142],[19,28518,28520],{"className":28519,"style":147},[146],[19,28521,28522,28525],{"style":150},[19,28523],{"className":28524,"style":155},[154],[19,28526,28528],{"className":28527},[159,160,161,162],[19,28529,28531],{"className":28530},[75,162],[19,28532,110],{"className":28533},[75,162],[19,28535,173],{"className":28536},[172],[19,28538,28540],{"className":28539},[142],[19,28541,28543],{"className":28542,"style":180},[146],[19,28544],{},", we obtain the general case mentioned above. Hence, the power series mentioned below all refer to the power series:",[19,28547,28549],{"className":28548,"translate":23},[187],[19,28550,28552,28592],{"className":28551,"translate":23},[22],[19,28553,28555],{"className":28554},[27],[29,28556,28557],{"xmlns":31,"display":196},[33,28558,28559,28589],{},[36,28560,28561,28575],{},[26840,28562,28563,28565,28573],{},[43,28564,26844],{},[36,28566,28567,28569,28571],{},[39,28568,3297],{},[43,28570,267],{},[108,28572,110],{},[39,28574,26855],{"mathvariant":275},[36,28576,28577,28583],{},[103,28578,28579,28581],{},[39,28580,3484],{},[39,28582,3297],{},[269,28584,28585,28587],{},[39,28586,49],{},[39,28588,3297],{},[54,28590,28591],{"encoding":56},"\\sum_{n=0}^{\\infty}{a_{n}x^{n}}",[19,28593,28595],{"className":28594,"ariaHidden":62},[61],[19,28596,28598,28601,28668,28671],{"className":28597},[66],[19,28599],{"className":28600,"style":26984},[70],[19,28602,28604],{"className":28603},[304,305],[19,28605,28607,28660],{"className":28606},[137,138],[19,28608,28610,28657],{"className":28609},[142],[19,28611,28613,28633,28643],{"className":28612,"style":26997},[146],[19,28614,28615,28618],{"style":27000},[19,28616],{"className":28617,"style":27004},[154],[19,28619,28621],{"className":28620},[159,160,161,162],[19,28622,28624,28627,28630],{"className":28623},[75,162],[19,28625,3297],{"className":28626},[75,76,162],[19,28628,267],{"className":28629},[335,162],[19,28631,110],{"className":28632},[75,162],[19,28634,28635,28638],{"style":27022},[19,28636],{"className":28637,"style":27004},[154],[19,28639,28640],{},[19,28641,26844],{"className":28642},[304,27031,27032],[19,28644,28645,28648],{"style":27035},[19,28646],{"className":28647,"style":27004},[154],[19,28649,28651],{"className":28650},[159,160,161,162],[19,28652,28654],{"className":28653},[75,162],[19,28655,26855],{"className":28656},[75,162],[19,28658,173],{"className":28659},[172],[19,28661,28663],{"className":28662},[142],[19,28664,28666],{"className":28665,"style":27057},[146],[19,28667],{},[19,28669],{"className":28670,"style":412},[411],[19,28672,28674,28717],{"className":28673},[75],[19,28675,28677,28680],{"className":28676},[75],[19,28678,3484],{"className":28679},[75,76],[19,28681,28683],{"className":28682},[133],[19,28684,28686,28709],{"className":28685},[137,138],[19,28687,28689,28706],{"className":28688},[142],[19,28690,28692],{"className":28691,"style":3609},[146],[19,28693,28694,28697],{"style":150},[19,28695],{"className":28696,"style":155},[154],[19,28698,28700],{"className":28699},[159,160,161,162],[19,28701,28703],{"className":28702},[75,162],[19,28704,3297],{"className":28705},[75,76,162],[19,28707,173],{"className":28708},[172],[19,28710,28712],{"className":28711},[142],[19,28713,28715],{"className":28714,"style":180},[146],[19,28716],{},[19,28718,28720,28723],{"className":28719},[75],[19,28721,49],{"className":28722},[75,76],[19,28724,28726],{"className":28725},[133],[19,28727,28729],{"className":28728},[137],[19,28730,28732],{"className":28731},[142],[19,28733,28735],{"className":28734,"style":4148},[146],[19,28736,28737,28740],{"style":648},[19,28738],{"className":28739,"style":155},[154],[19,28741,28743],{"className":28742},[159,160,161,162],[19,28744,28746],{"className":28745},[75,162],[19,28747,3297],{"className":28748},[75,76,162],[15,28750,28751],{},"When discussing a function series, the first thing we need to know is its domain of convergence. For a power series, its domain of convergence has a special characteristic, as illustrated in the following theorem:",[15,28753,28754,28757],{},[7600,28755,28756],{},"Abel's Theorem:"," If the power series",[19,28759,28761],{"className":28760,"translate":23},[187],[19,28762,28764,28803],{"className":28763,"translate":23},[22],[19,28765,28767],{"className":28766},[27],[29,28768,28769],{"xmlns":31,"display":196},[33,28770,28771,28801],{},[36,28772,28773,28787],{},[26840,28774,28775,28777,28785],{},[43,28776,26844],{},[36,28778,28779,28781,28783],{},[39,28780,3297],{},[43,28782,267],{},[108,28784,110],{},[39,28786,26855],{"mathvariant":275},[36,28788,28789,28795],{},[103,28790,28791,28793],{},[39,28792,3484],{},[39,28794,3297],{},[269,28796,28797,28799],{},[39,28798,49],{},[39,28800,3297],{},[54,28802,28591],{"encoding":56},[19,28804,28806],{"className":28805,"ariaHidden":62},[61],[19,28807,28809,28812,28879,28882],{"className":28808},[66],[19,28810],{"className":28811,"style":26984},[70],[19,28813,28815],{"className":28814},[304,305],[19,28816,28818,28871],{"className":28817},[137,138],[19,28819,28821,28868],{"className":28820},[142],[19,28822,28824,28844,28854],{"className":28823,"style":26997},[146],[19,28825,28826,28829],{"style":27000},[19,28827],{"className":28828,"style":27004},[154],[19,28830,28832],{"className":28831},[159,160,161,162],[19,28833,28835,28838,28841],{"className":28834},[75,162],[19,28836,3297],{"className":28837},[75,76,162],[19,28839,267],{"className":28840},[335,162],[19,28842,110],{"className":28843},[75,162],[19,28845,28846,28849],{"style":27022},[19,28847],{"className":28848,"style":27004},[154],[19,28850,28851],{},[19,28852,26844],{"className":28853},[304,27031,27032],[19,28855,28856,28859],{"style":27035},[19,28857],{"className":28858,"style":27004},[154],[19,28860,28862],{"className":28861},[159,160,161,162],[19,28863,28865],{"className":28864},[75,162],[19,28866,26855],{"className":28867},[75,162],[19,28869,173],{"className":28870},[172],[19,28872,28874],{"className":28873},[142],[19,28875,28877],{"className":28876,"style":27057},[146],[19,28878],{},[19,28880],{"className":28881,"style":412},[411],[19,28883,28885,28928],{"className":28884},[75],[19,28886,28888,28891],{"className":28887},[75],[19,28889,3484],{"className":28890},[75,76],[19,28892,28894],{"className":28893},[133],[19,28895,28897,28920],{"className":28896},[137,138],[19,28898,28900,28917],{"className":28899},[142],[19,28901,28903],{"className":28902,"style":3609},[146],[19,28904,28905,28908],{"style":150},[19,28906],{"className":28907,"style":155},[154],[19,28909,28911],{"className":28910},[159,160,161,162],[19,28912,28914],{"className":28913},[75,162],[19,28915,3297],{"className":28916},[75,76,162],[19,28918,173],{"className":28919},[172],[19,28921,28923],{"className":28922},[142],[19,28924,28926],{"className":28925,"style":180},[146],[19,28927],{},[19,28929,28931,28934],{"className":28930},[75],[19,28932,49],{"className":28933},[75,76],[19,28935,28937],{"className":28936},[133],[19,28938,28940],{"className":28939},[137],[19,28941,28943],{"className":28942},[142],[19,28944,28946],{"className":28945,"style":4148},[146],[19,28947,28948,28951],{"style":648},[19,28949],{"className":28950,"style":155},[154],[19,28952,28954],{"className":28953},[159,160,161,162],[19,28955,28957],{"className":28956},[75,162],[19,28958,3297],{"className":28959},[75,76,162],[15,28961,28962,28963,29115,29116,29144,29145,29260,29261,29115,29398,29144,29426,29540],{},"converges at ",[19,28964,28966,28995],{"className":28965,"translate":23},[22],[19,28967,28969],{"className":28968},[27],[29,28970,28971],{"xmlns":31},[33,28972,28973,28992],{},[36,28974,28975,28977,28979,28987,28990],{},[39,28976,49],{},[43,28978,267],{},[28980,28981,28982,28984],"mover",{"accent":62},[39,28983,49],{},[43,28985,28986],{},"ˉ",[43,28988,28989],{"mathvariant":275},"≠",[108,28991,110],{},[54,28993,28994],{"encoding":56},"x=\\bar{x}\\ne0",[19,28996,28998,29016,29106],{"className":28997,"ariaHidden":62},[61],[19,28999,29001,29004,29007,29010,29013],{"className":29000},[66],[19,29002],{"className":29003,"style":3309},[70],[19,29005,49],{"className":29006},[75,76],[19,29008],{"className":29009,"style":613},[411],[19,29011,267],{"className":29012},[335],[19,29014],{"className":29015,"style":613},[411],[19,29017,29019,29022,29057,29060,29103],{"className":29018},[66],[19,29020],{"className":29021,"style":6202},[70],[19,29023,29026],{"className":29024},[75,29025],"accent",[19,29027,29029],{"className":29028},[137],[19,29030,29032],{"className":29031},[142],[19,29033,29036,29044],{"className":29034,"style":29035},[146],"height:0.5678em;",[19,29037,29038,29041],{"style":387},[19,29039],{"className":29040,"style":322},[154],[19,29042,49],{"className":29043},[75,76],[19,29045,29046,29049],{"style":387},[19,29047],{"className":29048,"style":322},[154],[19,29050,29054],{"className":29051,"style":29053},[29052],"accent-body","left:-0.2222em;",[19,29055,28986],{"className":29056},[75],[19,29058],{"className":29059,"style":613},[411],[19,29061,29063,29096,29100],{"className":29062},[335],[19,29064,29066],{"className":29065},[335],[19,29067,29070],{"className":29068},[75,29069],"vbox",[19,29071,29074],{"className":29072},[29073],"thinbox",[19,29075,29078,29081,29092],{"className":29076},[29077],"rlap",[19,29079],{"className":29080,"style":6202},[70],[19,29082,29085],{"className":29083},[29084],"inner",[19,29086,29088],{"className":29087},[75],[19,29089,29091],{"className":29090},[335],"",[19,29093],{"className":29094},[29095],"fix",[19,29097],{"className":29098},[411,29099],"nobreak",[19,29101,267],{"className":29102},[335],[19,29104],{"className":29105,"style":613},[411],[19,29107,29109,29112],{"className":29108},[66],[19,29110],{"className":29111,"style":1214},[70],[19,29113,110],{"className":29114},[75],", then for any ",[19,29117,29119,29132],{"className":29118,"translate":23},[22],[19,29120,29122],{"className":29121},[27],[29,29123,29124],{"xmlns":31},[33,29125,29126,29130],{},[36,29127,29128],{},[39,29129,49],{},[54,29131,49],{"encoding":56},[19,29133,29135],{"className":29134,"ariaHidden":62},[61],[19,29136,29138,29141],{"className":29137},[66],[19,29139],{"className":29140,"style":3309},[70],[19,29142,49],{"className":29143},[75,76]," satisfying the inequality ",[19,29146,29148,29184],{"className":29147,"translate":23},[22],[19,29149,29151],{"className":29150},[27],[29,29152,29153],{"xmlns":31},[33,29154,29155,29181],{},[36,29156,29157,29166,29169],{},[36,29158,29159,29162,29164],{},[43,29160,29161],{"fence":62},"∣",[39,29163,49],{},[43,29165,29161],{"fence":62},[43,29167,29168],{},"\u003C",[36,29170,29171,29173,29179],{},[43,29172,29161],{"fence":62},[28980,29174,29175,29177],{"accent":62},[39,29176,49],{},[43,29178,28986],{},[43,29180,29161],{"fence":62},[54,29182,29183],{"encoding":56},"\\left| x \\right|\u003C\\left| \\bar{x} \\right|",[19,29185,29187,29214],{"className":29186,"ariaHidden":62},[61],[19,29188,29190,29193,29205,29208,29211],{"className":29189},[66],[19,29191],{"className":29192,"style":71},[70],[19,29194,29196,29199,29202],{"className":29195},[16026],[19,29197,29161],{"className":29198,"style":16031},[81,16030],[19,29200,49],{"className":29201},[75,76],[19,29203,29161],{"className":29204,"style":16031},[88,16030],[19,29206],{"className":29207,"style":613},[411],[19,29209,29168],{"className":29210},[335],[19,29212],{"className":29213,"style":613},[411],[19,29215,29217,29220],{"className":29216},[66],[19,29218],{"className":29219,"style":71},[70],[19,29221,29223,29226,29257],{"className":29222},[16026],[19,29224,29161],{"className":29225,"style":16031},[81,16030],[19,29227,29229],{"className":29228},[75,29025],[19,29230,29232],{"className":29231},[137],[19,29233,29235],{"className":29234},[142],[19,29236,29238,29246],{"className":29237,"style":29035},[146],[19,29239,29240,29243],{"style":387},[19,29241],{"className":29242,"style":322},[154],[19,29244,49],{"className":29245},[75,76],[19,29247,29248,29251],{"style":387},[19,29249],{"className":29250,"style":322},[154],[19,29252,29254],{"className":29253,"style":29053},[29052],[19,29255,28986],{"className":29256},[75],[19,29258,29161],{"className":29259,"style":16031},[88,16030],", the power series converges and converges absolutely. If the power series diverges at ",[19,29262,29264,29289],{"className":29263,"translate":23},[22],[19,29265,29267],{"className":29266},[27],[29,29268,29269],{"xmlns":31},[33,29270,29271,29287],{},[36,29272,29273,29275,29277,29283,29285],{},[39,29274,49],{},[43,29276,267],{},[28980,29278,29279,29281],{"accent":62},[39,29280,49],{},[43,29282,28986],{},[43,29284,28989],{"mathvariant":275},[108,29286,110],{},[54,29288,28994],{"encoding":56},[19,29290,29292,29310,29389],{"className":29291,"ariaHidden":62},[61],[19,29293,29295,29298,29301,29304,29307],{"className":29294},[66],[19,29296],{"className":29297,"style":3309},[70],[19,29299,49],{"className":29300},[75,76],[19,29302],{"className":29303,"style":613},[411],[19,29305,267],{"className":29306},[335],[19,29308],{"className":29309,"style":613},[411],[19,29311,29313,29316,29347,29350,29386],{"className":29312},[66],[19,29314],{"className":29315,"style":6202},[70],[19,29317,29319],{"className":29318},[75,29025],[19,29320,29322],{"className":29321},[137],[19,29323,29325],{"className":29324},[142],[19,29326,29328,29336],{"className":29327,"style":29035},[146],[19,29329,29330,29333],{"style":387},[19,29331],{"className":29332,"style":322},[154],[19,29334,49],{"className":29335},[75,76],[19,29337,29338,29341],{"style":387},[19,29339],{"className":29340,"style":322},[154],[19,29342,29344],{"className":29343,"style":29053},[29052],[19,29345,28986],{"className":29346},[75],[19,29348],{"className":29349,"style":613},[411],[19,29351,29353,29380,29383],{"className":29352},[335],[19,29354,29356],{"className":29355},[335],[19,29357,29359],{"className":29358},[75,29069],[19,29360,29362],{"className":29361},[29073],[19,29363,29365,29368,29377],{"className":29364},[29077],[19,29366],{"className":29367,"style":6202},[70],[19,29369,29371],{"className":29370},[29084],[19,29372,29374],{"className":29373},[75],[19,29375,29091],{"className":29376},[335],[19,29378],{"className":29379},[29095],[19,29381],{"className":29382},[411,29099],[19,29384,267],{"className":29385},[335],[19,29387],{"className":29388,"style":613},[411],[19,29390,29392,29395],{"className":29391},[66],[19,29393],{"className":29394,"style":1214},[70],[19,29396,110],{"className":29397},[75],[19,29399,29401,29414],{"className":29400,"translate":23},[22],[19,29402,29404],{"className":29403},[27],[29,29405,29406],{"xmlns":31},[33,29407,29408,29412],{},[36,29409,29410],{},[39,29411,49],{},[54,29413,49],{"encoding":56},[19,29415,29417],{"className":29416,"ariaHidden":62},[61],[19,29418,29420,29423],{"className":29419},[66],[19,29421],{"className":29422,"style":3309},[70],[19,29424,49],{"className":29425},[75,76],[19,29427,29429,29464],{"className":29428,"translate":23},[22],[19,29430,29432],{"className":29431},[27],[29,29433,29434],{"xmlns":31},[33,29435,29436,29461],{},[36,29437,29438,29446,29449],{},[36,29439,29440,29442,29444],{},[43,29441,29161],{"fence":62},[39,29443,49],{},[43,29445,29161],{"fence":62},[43,29447,29448],{},">",[36,29450,29451,29453,29459],{},[43,29452,29161],{"fence":62},[28980,29454,29455,29457],{"accent":62},[39,29456,49],{},[43,29458,28986],{},[43,29460,29161],{"fence":62},[54,29462,29463],{"encoding":56},"\\left| x \\right|>\\left| \\bar{x} \\right|",[19,29465,29467,29494],{"className":29466,"ariaHidden":62},[61],[19,29468,29470,29473,29485,29488,29491],{"className":29469},[66],[19,29471],{"className":29472,"style":71},[70],[19,29474,29476,29479,29482],{"className":29475},[16026],[19,29477,29161],{"className":29478,"style":16031},[81,16030],[19,29480,49],{"className":29481},[75,76],[19,29483,29161],{"className":29484,"style":16031},[88,16030],[19,29486],{"className":29487,"style":613},[411],[19,29489,29448],{"className":29490},[335],[19,29492],{"className":29493,"style":613},[411],[19,29495,29497,29500],{"className":29496},[66],[19,29498],{"className":29499,"style":71},[70],[19,29501,29503,29506,29537],{"className":29502},[16026],[19,29504,29161],{"className":29505,"style":16031},[81,16030],[19,29507,29509],{"className":29508},[75,29025],[19,29510,29512],{"className":29511},[137],[19,29513,29515],{"className":29514},[142],[19,29516,29518,29526],{"className":29517,"style":29035},[146],[19,29519,29520,29523],{"style":387},[19,29521],{"className":29522,"style":322},[154],[19,29524,49],{"className":29525},[75,76],[19,29527,29528,29531],{"style":387},[19,29529],{"className":29530,"style":322},[154],[19,29532,29534],{"className":29533,"style":29053},[29052],[19,29535,28986],{"className":29536},[75],[19,29538,29161],{"className":29539,"style":16031},[88,16030],", the power series diverges.",[15,29542,29543,29545],{},[7600,29544,10737],{}," Assume the 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By the necessary condition for the convergence of series, the sequence ",[19,29785,29787,29819],{"className":29786,"translate":23},[22],[19,29788,29790],{"className":29789},[27],[29,29791,29792],{"xmlns":31},[33,29793,29794,29816],{},[36,29795,29796,29798,29804,29814],{},[43,29797,26641],{"fence":62},[103,29799,29800,29802],{},[39,29801,3484],{},[39,29803,3297],{},[269,29805,29806,29812],{},[28980,29807,29808,29810],{"accent":62},[39,29809,49],{},[43,29811,28986],{},[39,29813,3297],{},[43,29815,26668],{"fence":62},[54,29817,29818],{"encoding":56},"\\left\\{ a_{n}\\bar{x}^{n} \\right\\}",[19,29820,29822],{"className":29821,"ariaHidden":62},[61],[19,29823,29825,29828],{"className":29824},[66],[19,29826],{"className":29827,"style":71},[70],[19,29829,29831,29834,29877,29937],{"className":29830},[16026],[19,29832,26641],{"className":29833,"style":16031},[81,16030],[19,29835,29837,29840],{"className":29836},[75],[19,29838,3484],{"className":29839},[75,76],[19,29841,29843],{"className":29842},[133],[19,29844,29846,29869],{"className":29845},[137,138],[19,29847,29849,29866],{"className":29848},[142],[19,29850,29852],{"className":29851,"style":3609},[146],[19,29853,29854,29857],{"style":150},[19,29855],{"className":29856,"style":155},[154],[19,29858,29860],{"className":29859},[159,160,161,162],[19,29861,29863],{"className":29862},[75,162],[19,29864,3297],{"className":29865},[75,76,162],[19,29867,173],{"className":29868},[172],[19,29870,29872],{"className":29871},[142],[19,29873,29875],{"className":29874,"style":180},[146],[19,29876],{},[19,29878,29880,29911],{"className":29879},[75],[19,29881,29883],{"className":29882},[75,29025],[19,29884,29886],{"className":29885},[137],[19,29887,29889],{"className":29888},[142],[19,29890,29892,29900],{"className":29891,"style":29035},[146],[19,29893,29894,29897],{"style":387},[19,29895],{"className":29896,"style":322},[154],[19,29898,49],{"className":29899},[75,76],[19,29901,29902,29905],{"style":387},[19,29903],{"className":29904,"style":322},[154],[19,29906,29908],{"className":29907,"style":29053},[29052],[19,29909,28986],{"className":29910},[75],[19,29912,29914],{"className":29913},[133],[19,29915,29917],{"className":29916},[137],[19,29918,29920],{"className":29919},[142],[19,29921,29923],{"className":29922,"style":3429},[146],[19,29924,29925,29928],{"style":1643},[19,29926],{"className":29927,"style":155},[154],[19,29929,29931],{"className":29930},[159,160,161,162],[19,29932,29934],{"className":29933},[75,162],[19,29935,3297],{"className":29936},[75,76,162],[19,29938,26668],{"className":29939,"style":16031},[88,16030]," converges to zero and is bounded. Hence, there exists a positive number ",[19,29942,29944,29958],{"className":29943,"translate":23},[22],[19,29945,29947],{"className":29946},[27],[29,29948,29949],{"xmlns":31},[33,29950,29951,29956],{},[36,29952,29953],{},[39,29954,29955],{},"M",[54,29957,29955],{"encoding":56},[19,29959,29961],{"className":29960,"ariaHidden":62},[61],[19,29962,29964,29967],{"className":29963},[66],[19,29965],{"className":29966,"style":7738},[70],[19,29968,29955],{"className":29969,"style":13222},[75,76],[19,29971,29973],{"className":29972,"translate":23},[187],[19,29974,29976,30042],{"className":29975,"translate":23},[22],[19,29977,29979],{"className":29978},[27],[29,29980,29981],{"xmlns":31,"display":196},[33,29982,29983,30039],{},[36,29984,29985,30007,30009,30011,30014,30016,30018,30020,30022,30024,30026,30028,30030,30032,30034,30037],{},[36,29986,29987,29989,29995,30005],{},[43,29988,29161],{"fence":62},[103,29990,29991,29993],{},[39,29992,3484],{},[39,29994,3297],{},[269,29996,29997,30003],{},[28980,29998,29999,30001],{"accent":62},[39,30000,49],{},[43,30002,28986],{},[39,30004,3297],{},[43,30006,29161],{"fence":62},[43,30008,29168],{},[39,30010,29955],{},[411,30012],{"width":30013},"1em",[43,30015,46],{"stretchy":45},[39,30017,3297],{},[43,30019,267],{},[108,30021,110],{},[43,30023,7830],{"separator":62},[108,30025,3495],{},[43,30027,7830],{"separator":62},[108,30029,3518],{},[43,30031,7830],{"separator":62},[43,30033,21943],{},[8333,30035,30036],{}," ",[43,30038,52],{"stretchy":45},[54,30040,30041],{"encoding":56},"\\left| 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h43z",[19,31402,173],{"className":31403},[172],[19,31405,31407],{"className":31406},[142],[19,31408,31411],{"className":31409,"style":31410},[146],"height:0.65em;",[19,31412],{},[19,31414,31416,31419,31501],{"className":31415},[75],[19,31417],{"className":31418},[81,422],[19,31420,31422],{"className":31421},[226],[19,31423,31425,31493],{"className":31424},[137,138],[19,31426,31428,31490],{"className":31427},[142],[19,31429,31432,31471,31479],{"className":31430,"style":31431},[146],"height:1.1076em;",[19,31433,31434,31437],{"style":438},[19,31435],{"className":31436,"style":322},[154],[19,31438,31440],{"className":31439},[75],[19,31441,31443],{"className":31442},[75,29025],[19,31444,31446],{"className":31445},[137],[19,31447,31449],{"className":31448},[142],[19,31450,31452,31460],{"className":31451,"style":29035},[146],[19,31453,31454,31457],{"style":387},[19,31455],{"className":31456,"style":322},[154],[19,31458,49],{"className":31459},[75,76],[19,31461,31462,31465],{"style":387},[19,31463],{"className":31464,"style":322},[154],[19,31466,31468],{"className":31467,"style":29053},[29052],[19,31469,28986],{"className":31470},[75],[19,31472,31473,31476],{"style":504},[19,31474],{"className":31475,"style":322},[154],[19,31477],{"className":31478,"style":512},[511],[19,31480,31481,31484],{"style":515},[19,31482],{"className":31483,"style":322},[154],[19,31485,31487],{"className":31486},[75],[19,31488,49],{"className":31489},[75,76],[19,31491,173],{"className":31492},[172],[19,31494,31496],{"className":31495},[142],[19,31497,31499],{"className":31498,"style":5671},[146],[19,31500],{},[19,31502],{"className":31503},[88,422],[19,31505,31507],{"className":31506},[88],[19,31508,31510],{"className":31509},[16035,30439],[19,31511,31513,31533],{"className":31512},[137,138],[19,31514,31516,31530],{"className":31515},[142],[19,31517,31519],{"className":31518,"style":31383},[146],[19,31520,31521,31524],{"style":31386},[19,31522],{"className":31523,"style":31390},[154],[19,31525,31526],{"style":31393},[30461,31527,31528],{"xmlns":30463,"width":30464,"height":31396,"viewBox":31397},[30468,31529],{"d":31400},[19,31531,173],{"className":31532},[172],[19,31534,31536],{"className":31535},[142],[19,31537,31539],{"className":31538,"style":31410},[146],[19,31540],{},[19,31542,31544],{"className":31543},[133],[19,31545,31547],{"className":31546},[137],[19,31548,31550],{"className":31549},[142],[19,31551,31554],{"className":31552,"style":31553},[146],"height:1.2043em;",[19,31555,31557,31560],{"style":31556},"top:-3.6029em;margin-right:0.05em;",[19,31558],{"className":31559,"style":155},[154],[19,31561,31563],{"className":31562},[159,160,161,162],[19,31564,31566],{"className":31565},[75,162],[19,31567,3297],{"className":31568},[75,76,162],[19,31570],{"className":31571,"style":613},[411],[19,31573,29168],{"className":31574},[335],[19,31576],{"className":31577,"style":613},[411],[19,31579,31581,31584,31587],{"className":31580},[66],[19,31582],{"className":31583,"style":4148},[70],[19,31585,29955],{"className":31586,"style":13222},[75,76],[19,31588,31590,31594],{"className":31589},[75],[19,31591,30769],{"className":31592,"style":31593},[75,76],"margin-right:0.02778em;",[19,31595,31597],{"className":31596},[133],[19,31598,31600],{"className":31599},[137],[19,31601,31603],{"className":31602},[142],[19,31604,31606],{"className":31605,"style":4148},[146],[19,31607,31608,31611],{"style":648},[19,31609],{"className":31610,"style":155},[154],[19,31612,31614],{"className":31613},[159,160,161,162],[19,31615,31617],{"className":31616},[75,162],[19,31618,3297],{"className":31619},[75,76,162],[15,31621,31622,31623,31880,31881,32040,32041,32236,32237,3447],{},"where ",[19,31624,31626,31662],{"className":31625,"translate":23},[22],[19,31627,31629],{"className":31628},[27],[29,31630,31631],{"xmlns":31},[33,31632,31633,31659],{},[36,31634,31635,31637,31639,31655,31657],{},[39,31636,30769],{},[43,31638,267],{},[36,31640,31641,31643,31653],{},[43,31642,29161],{"fence":62},[226,31644,31645,31647],{},[39,31646,49],{},[28980,31648,31649,31651],{"accent":62},[39,31650,49],{},[43,31652,28986],{},[43,31654,29161],{"fence":62},[43,31656,29168],{},[108,31658,3495],{},[54,31660,31661],{"encoding":56},"r=\\left| \\frac{x}{\\bar{x}} \\right|\u003C1",[19,31663,31665,31683,31871],{"className":31664,"ariaHidden":62},[61],[19,31666,31668,31671,31674,31677,31680],{"className":31667},[66],[19,31669],{"className":31670,"style":3309},[70],[19,31672,30769],{"className":31673,"style":31593},[75,76],[19,31675],{"className":31676,"style":613},[411],[19,31678,267],{"className":31679},[335],[19,31681],{"className":31682,"style":613},[411],[19,31684,31686,31689,31862,31865,31868],{"className":31685},[66],[19,31687],{"className":31688,"style":30429},[70],[19,31690,31692,31729,31825],{"className":31691},[16026],[19,31693,31695],{"className":31694},[81],[19,31696,31698],{"className":31697},[16035,30439],[19,31699,31701,31721],{"className":31700},[137,138],[19,31702,31704,31718],{"className":31703},[142],[19,31705,31707],{"className":31706,"style":30449},[146],[19,31708,31709,31712],{"style":30452},[19,31710],{"className":31711,"style":30456},[154],[19,31713,31714],{"style":30459},[30461,31715,31716],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,31717],{"d":30470},[19,31719,173],{"className":31720},[172],[19,31722,31724],{"className":31723},[142],[19,31725,31727],{"className":31726,"style":30480},[146],[19,31728],{},[19,31730,31732,31735,31822],{"className":31731},[75],[19,31733],{"className":31734},[81,422],[19,31736,31738],{"className":31737},[226],[19,31739,31741,31814],{"className":31740},[137,138],[19,31742,31744,31811],{"className":31743},[142],[19,31745,31747,31789,31797],{"className":31746,"style":30501},[146],[19,31748,31749,31752],{"style":30504},[19,31750],{"className":31751,"style":322},[154],[19,31753,31755],{"className":31754},[159,160,161,162],[19,31756,31758],{"className":31757},[75,162],[19,31759,31761],{"className":31760},[75,29025,162],[19,31762,31764],{"className":31763},[137],[19,31765,31767],{"className":31766},[142],[19,31768,31770,31778],{"className":31769,"style":29035},[146],[19,31771,31772,31775],{"style":30528},[19,31773],{"className":31774,"style":155},[154],[19,31776,49],{"className":31777},[75,76,162],[19,31779,31780,31783],{"style":30528},[19,31781],{"className":31782,"style":155},[154],[19,31784,31786],{"className":31785,"style":29053},[29052],[19,31787,28986],{"className":31788},[75,162],[19,31790,31791,31794],{"style":504},[19,31792],{"className":31793,"style":322},[154],[19,31795],{"className":31796,"style":512},[511],[19,31798,31799,31802],{"style":30556},[19,31800],{"className":31801,"style":322},[154],[19,31803,31805],{"className":31804},[159,160,161,162],[19,31806,31808],{"className":31807},[75,162],[19,31809,49],{"className":31810},[75,76,162],[19,31812,173],{"className":31813},[172],[19,31815,31817],{"className":31816},[142],[19,31818,31820],{"className":31819,"style":30578},[146],[19,31821],{},[19,31823],{"className":31824},[88,422],[19,31826,31828],{"className":31827},[88],[19,31829,31831],{"className":31830},[16035,30439],[19,31832,31834,31854],{"className":31833},[137,138],[19,31835,31837,31851],{"className":31836},[142],[19,31838,31840],{"className":31839,"style":30449},[146],[19,31841,31842,31845],{"style":30452},[19,31843],{"className":31844,"style":30456},[154],[19,31846,31847],{"style":30459},[30461,31848,31849],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,31850],{"d":30470},[19,31852,173],{"className":31853},[172],[19,31855,31857],{"className":31856},[142],[19,31858,31860],{"className":31859,"style":30480},[146],[19,31861],{},[19,31863],{"className":31864,"style":613},[411],[19,31866,29168],{"className":31867},[335],[19,31869],{"className":31870,"style":613},[411],[19,31872,31874,31877],{"className":31873},[66],[19,31875],{"className":31876,"style":1214},[70],[19,31878,3495],{"className":31879},[75],". Since the series ",[19,31882,31884,31920],{"className":31883,"translate":23},[22],[19,31885,31887],{"className":31886},[27],[29,31888,31889],{"xmlns":31},[33,31890,31891,31917],{},[36,31892,31893,31907],{},[4285,31894,31895,31897,31905],{},[43,31896,26844],{},[36,31898,31899,31901,31903],{},[39,31900,3297],{},[43,31902,267],{},[108,31904,110],{},[39,31906,26855],{"mathvariant":275},[36,31908,31909,31911],{},[39,31910,29955],{},[269,31912,31913,31915],{},[39,31914,30769],{},[39,31916,3297],{},[54,31918,31919],{"encoding":56},"\\sum_{n=0}^{\\infty}{Mr^{n}}",[19,31921,31923],{"className":31922,"ariaHidden":62},[61],[19,31924,31926,31930,31999,32002],{"className":31925},[66],[19,31927],{"className":31928,"style":31929},[70],"height:1.104em;vertical-align:-0.2997em;",[19,31931,31933,31938],{"className":31932},[304],[19,31934,26844],{"className":31935,"style":31937},[304,27031,31936],"small-op","position:relative;top:0em;",[19,31939,31941],{"className":31940},[133],[19,31942,31944,31990],{"className":31943},[137,138],[19,31945,31947,31987],{"className":31946},[142],[19,31948,31951,31972],{"className":31949,"style":31950},[146],"height:0.8043em;",[19,31952,31954,31957],{"style":31953},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[19,31955],{"className":31956,"style":155},[154],[19,31958,31960],{"className":31959},[159,160,161,162],[19,31961,31963,31966,31969],{"className":31962},[75,162],[19,31964,3297],{"className":31965},[75,76,162],[19,31967,267],{"className":31968},[335,162],[19,31970,110],{"className":31971},[75,162],[19,31973,31975,31978],{"style":31974},"top:-3.2029em;margin-right:0.05em;",[19,31976],{"className":31977,"style":155},[154],[19,31979,31981],{"className":31980},[159,160,161,162],[19,31982,31984],{"className":31983},[75,162],[19,31985,26855],{"className":31986},[75,162],[19,31988,173],{"className":31989},[172],[19,31991,31993],{"className":31992},[142],[19,31994,31997],{"className":31995,"style":31996},[146],"height:0.2997em;",[19,31998],{},[19,32000],{"className":32001,"style":412},[411],[19,32003,32005,32008],{"className":32004},[75],[19,32006,29955],{"className":32007,"style":13222},[75,76],[19,32009,32011,32014],{"className":32010},[75],[19,32012,30769],{"className":32013,"style":31593},[75,76],[19,32015,32017],{"className":32016},[133],[19,32018,32020],{"className":32019},[137],[19,32021,32023],{"className":32022},[142],[19,32024,32026],{"className":32025,"style":3429},[146],[19,32027,32028,32031],{"style":1643},[19,32029],{"className":32030,"style":155},[154],[19,32032,32034],{"className":32033},[159,160,161,162],[19,32035,32037],{"className":32036},[75,162],[19,32038,3297],{"className":32039},[75,76,162]," converges, the power series 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converges absolutely when ",[19,32238,32240,32273],{"className":32239,"translate":23},[22],[19,32241,32243],{"className":32242},[27],[29,32244,32245],{"xmlns":31},[33,32246,32247,32271],{},[36,32248,32249,32257,32259],{},[36,32250,32251,32253,32255],{},[43,32252,29161],{"fence":62},[39,32254,49],{},[43,32256,29161],{"fence":62},[43,32258,29168],{},[36,32260,32261,32263,32269],{},[43,32262,29161],{"fence":62},[28980,32264,32265,32267],{"accent":62},[39,32266,49],{},[43,32268,28986],{},[43,32270,29161],{"fence":62},[54,32272,29183],{"encoding":56},[19,32274,32276,32303],{"className":32275,"ariaHidden":62},[61],[19,32277,32279,32282,32294,32297,32300],{"className":32278},[66],[19,32280],{"className":32281,"style":71},[70],[19,32283,32285,32288,32291],{"className":32284},[16026],[19,32286,29161],{"className":32287,"style":16031},[81,16030],[19,32289,49],{"className":32290},[75,76],[19,32292,29161],{"className":32293,"style":16031},[88,16030],[19,32295],{"className":32296,"style":613},[411],[19,32298,29168],{"className":32299},[335],[19,32301],{"className":32302,"style":613},[411],[19,32304,32306,32309],{"className":32305},[66],[19,32307],{"className":32308,"style":71},[70],[19,32310,32312,32315,32346],{"className":32311},[16026],[19,32313,29161],{"className":32314,"style":16031},[81,16030],[19,32316,32318],{"className":32317},[75,29025],[19,32319,32321],{"className":32320},[137],[19,32322,32324],{"className":32323},[142],[19,32325,32327,32335],{"className":32326,"style":29035},[146],[19,32328,32329,32332],{"style":387},[19,32330],{"className":32331,"style":322},[154],[19,32333,49],{"className":32334},[75,76],[19,32336,32337,32340],{"style":387},[19,32338],{"className":32339,"style":322},[154],[19,32341,32343],{"className":32342,"style":29053},[29052],[19,32344,28986],{"className":32345},[75],[19,32347,29161],{"className":32348,"style":16031},[88,16030],[15,32350,32351,32352,32489,32490,32562,32563,32720,32721,32946,32947,33030,33031,29144,33059,33171,33172,33399],{},"Conversely, if the power series diverges at ",[19,32353,32355,32380],{"className":32354,"translate":23},[22],[19,32356,32358],{"className":32357},[27],[29,32359,32360],{"xmlns":31},[33,32361,32362,32378],{},[36,32363,32364,32366,32368,32374,32376],{},[39,32365,49],{},[43,32367,267],{},[28980,32369,32370,32372],{"accent":62},[39,32371,49],{},[43,32373,28986],{},[43,32375,28989],{"mathvariant":275},[108,32377,110],{},[54,32379,28994],{"encoding":56},[19,32381,32383,32401,32480],{"className":32382,"ariaHidden":62},[61],[19,32384,32386,32389,32392,32395,32398],{"className":32385},[66],[19,32387],{"className":32388,"style":3309},[70],[19,32390,49],{"className":32391},[75,76],[19,32393],{"className":32394,"style":613},[411],[19,32396,267],{"className":32397},[335],[19,32399],{"className":32400,"style":613},[411],[19,32402,32404,32407,32438,32441,32477],{"className":32403},[66],[19,32405],{"className":32406,"style":6202},[70],[19,32408,32410],{"className":32409},[75,29025],[19,32411,32413],{"className":32412},[137],[19,32414,32416],{"className":32415},[142],[19,32417,32419,32427],{"className":32418,"style":29035},[146],[19,32420,32421,32424],{"style":387},[19,32422],{"className":32423,"style":322},[154],[19,32425,49],{"className":32426},[75,76],[19,32428,32429,32432],{"style":387},[19,32430],{"className":32431,"style":322},[154],[19,32433,32435],{"className":32434,"style":29053},[29052],[19,32436,28986],{"className":32437},[75],[19,32439],{"className":32440,"style":613},[411],[19,32442,32444,32471,32474],{"className":32443},[335],[19,32445,32447],{"className":32446},[335],[19,32448,32450],{"className":32449},[75,29069],[19,32451,32453],{"className":32452},[29073],[19,32454,32456,32459,32468],{"className":32455},[29077],[19,32457],{"className":32458,"style":6202},[70],[19,32460,32462],{"className":32461},[29084],[19,32463,32465],{"className":32464},[75],[19,32466,29091],{"className":32467},[335],[19,32469],{"className":32470},[29095],[19,32472],{"className":32473},[411,29099],[19,32475,267],{"className":32476},[335],[19,32478],{"className":32479,"style":613},[411],[19,32481,32483,32486],{"className":32482},[66],[19,32484],{"className":32485,"style":1214},[70],[19,32487,110],{"className":32488},[75],", and if there exists ",[19,32491,32493,32510],{"className":32492,"translate":23},[22],[19,32494,32496],{"className":32495},[27],[29,32497,32498],{"xmlns":31},[33,32499,32500,32508],{},[36,32501,32502],{},[103,32503,32504,32506],{},[39,32505,49],{},[108,32507,110],{},[54,32509,113],{"encoding":56},[19,32511,32513],{"className":32512,"ariaHidden":62},[61],[19,32514,32516,32519],{"className":32515},[66],[19,32517],{"className":32518,"style":123},[70],[19,32520,32522,32525],{"className":32521},[75],[19,32523,49],{"className":32524},[75,76],[19,32526,32528],{"className":32527},[133],[19,32529,32531,32554],{"className":32530},[137,138],[19,32532,32534,32551],{"className":32533},[142],[19,32535,32537],{"className":32536,"style":147},[146],[19,32538,32539,32542],{"style":150},[19,32540],{"className":32541,"style":155},[154],[19,32543,32545],{"className":32544},[159,160,161,162],[19,32546,32548],{"className":32547},[75,162],[19,32549,110],{"className":32550},[75,162],[19,32552,173],{"className":32553},[172],[19,32555,32557],{"className":32556},[142],[19,32558,32560],{"className":32559,"style":180},[146],[19,32561],{}," such that ",[19,32564,32566,32604],{"className":32565,"translate":23},[22],[19,32567,32569],{"className":32568},[27],[29,32570,32571],{"xmlns":31},[33,32572,32573,32601],{},[36,32574,32575,32587,32589],{},[36,32576,32577,32579,32585],{},[43,32578,29161],{"fence":62},[103,32580,32581,32583],{},[39,32582,49],{},[108,32584,110],{},[43,32586,29161],{"fence":62},[43,32588,29448],{},[36,32590,32591,32593,32599],{},[43,32592,29161],{"fence":62},[28980,32594,32595,32597],{"accent":62},[39,32596,49],{},[43,32598,28986],{},[43,32600,29161],{"fence":62},[54,32602,32603],{"encoding":56},"\\left| x_{0} \\right|>\\left| \\bar{x} \\right|",[19,32605,32607,32674],{"className":32606,"ariaHidden":62},[61],[19,32608,32610,32613,32665,32668,32671],{"className":32609},[66],[19,32611],{"className":32612,"style":71},[70],[19,32614,32616,32619,32662],{"className":32615},[16026],[19,32617,29161],{"className":32618,"style":16031},[81,16030],[19,32620,32622,32625],{"className":32621},[75],[19,32623,49],{"className":32624},[75,76],[19,32626,32628],{"className":32627},[133],[19,32629,32631,32654],{"className":32630},[137,138],[19,32632,32634,32651],{"className":32633},[142],[19,32635,32637],{"className":32636,"style":147},[146],[19,32638,32639,32642],{"style":150},[19,32640],{"className":32641,"style":155},[154],[19,32643,32645],{"className":32644},[159,160,161,162],[19,32646,32648],{"className":32647},[75,162],[19,32649,110],{"className":32650},[75,162],[19,32652,173],{"className":32653},[172],[19,32655,32657],{"className":32656},[142],[19,32658,32660],{"className":32659,"style":180},[146],[19,32661],{},[19,32663,29161],{"className":32664,"style":16031},[88,16030],[19,32666],{"className":32667,"style":613},[411],[19,32669,29448],{"className":32670},[335],[19,32672],{"className":32673,"style":613},[411],[19,32675,32677,32680],{"className":32676},[66],[19,32678],{"className":32679,"style":71},[70],[19,32681,32683,32686,32717],{"className":32682},[16026],[19,32684,29161],{"className":32685,"style":16031},[81,16030],[19,32687,32689],{"className":32688},[75,29025],[19,32690,32692],{"className":32691},[137],[19,32693,32695],{"className":32694},[142],[19,32696,32698,32706],{"className":32697,"style":29035},[146],[19,32699,32700,32703],{"style":387},[19,32701],{"className":32702,"style":322},[154],[19,32704,49],{"className":32705},[75,76],[19,32707,32708,32711],{"style":387},[19,32709],{"className":32710,"style":322},[154],[19,32712,32714],{"className":32713,"style":29053},[29052],[19,32715,28986],{"className":32716},[75],[19,32718,29161],{"className":32719,"style":16031},[88,16030]," and the series ",[19,32722,32724,32766],{"className":32723,"translate":23},[22],[19,32725,32727],{"className":32726},[27],[29,32728,32729],{"xmlns":31},[33,32730,32731,32763],{},[36,32732,32733,32747],{},[4285,32734,32735,32737,32745],{},[43,32736,26844],{},[36,32738,32739,32741,32743],{},[39,32740,3297],{},[43,32742,267],{},[108,32744,110],{},[39,32746,26855],{"mathvariant":275},[36,32748,32749,32755],{},[103,32750,32751,32753],{},[39,32752,3484],{},[39,32754,3297],{},[4285,32756,32757,32759,32761],{},[39,32758,49],{},[108,32760,110],{},[39,32762,3297],{},[54,32764,32765],{"encoding":56},"\\sum_{n=0}^{\\infty}{a_{n}x_{0}^{n}}",[19,32767,32769],{"className":32768,"ariaHidden":62},[61],[19,32770,32772,32775,32838,32841],{"className":32771},[66],[19,32773],{"className":32774,"style":31929},[70],[19,32776,32778,32781],{"className":32777},[304],[19,32779,26844],{"className":32780,"style":31937},[304,27031,31936],[19,32782,32784],{"className":32783},[133],[19,32785,32787,32830],{"className":32786},[137,138],[19,32788,32790,32827],{"className":32789},[142],[19,32791,32793,32813],{"className":32792,"style":31950},[146],[19,32794,32795,32798],{"style":31953},[19,32796],{"className":32797,"style":155},[154],[19,32799,32801],{"className":32800},[159,160,161,162],[19,32802,32804,32807,32810],{"className":32803},[75,162],[19,32805,3297],{"className":32806},[75,76,162],[19,32808,267],{"className":32809},[335,162],[19,32811,110],{"className":32812},[75,162],[19,32814,32815,32818],{"style":31974},[19,32816],{"className":32817,"style":155},[154],[19,32819,32821],{"className":32820},[159,160,161,162],[19,32822,32824],{"className":32823},[75,162],[19,32825,26855],{"className":32826},[75,162],[19,32828,173],{"className":32829},[172],[19,32831,32833],{"className":32832},[142],[19,32834,32836],{"className":32835,"style":31996},[146],[19,32837],{},[19,32839],{"className":32840,"style":412},[411],[19,32842,32844,32887],{"className":32843},[75],[19,32845,32847,32850],{"className":32846},[75],[19,32848,3484],{"className":32849},[75,76],[19,32851,32853],{"className":32852},[133],[19,32854,32856,32879],{"className":32855},[137,138],[19,32857,32859,32876],{"className":32858},[142],[19,32860,32862],{"className":32861,"style":3609},[146],[19,32863,32864,32867],{"style":150},[19,32865],{"className":32866,"style":155},[154],[19,32868,32870],{"className":32869},[159,160,161,162],[19,32871,32873],{"className":32872},[75,162],[19,32874,3297],{"className":32875},[75,76,162],[19,32877,173],{"className":32878},[172],[19,32880,32882],{"className":32881},[142],[19,32883,32885],{"className":32884,"style":180},[146],[19,32886],{},[19,32888,32890,32893],{"className":32889},[75],[19,32891,49],{"className":32892},[75,76],[19,32894,32896],{"className":32895},[133],[19,32897,32899,32937],{"className":32898},[137,138],[19,32900,32902,32934],{"className":32901},[142],[19,32903,32905,32920],{"className":32904,"style":3429},[146],[19,32906,32908,32911],{"style":32907},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[19,32909],{"className":32910,"style":155},[154],[19,32912,32914],{"className":32913},[159,160,161,162],[19,32915,32917],{"className":32916},[75,162],[19,32918,110],{"className":32919},[75,162],[19,32921,32922,32925],{"style":1643},[19,32923],{"className":32924,"style":155},[154],[19,32926,32928],{"className":32927},[159,160,161,162],[19,32929,32931],{"className":32930},[75,162],[19,32932,3297],{"className":32933},[75,76,162],[19,32935,173],{"className":32936},[172],[19,32938,32940],{"className":32939},[142],[19,32941,32944],{"className":32942,"style":32943},[146],"height:0.2481em;",[19,32945],{}," converges, then according to the previous conclusion, the power series should converge absolutely at ",[19,32948,32950,32972],{"className":32949,"translate":23},[22],[19,32951,32953],{"className":32952},[27],[29,32954,32955],{"xmlns":31},[33,32956,32957,32969],{},[36,32958,32959,32961,32963],{},[39,32960,49],{},[43,32962,267],{},[28980,32964,32965,32967],{"accent":62},[39,32966,49],{},[43,32968,28986],{},[54,32970,32971],{"encoding":56},"x=\\bar{x}",[19,32973,32975,32993],{"className":32974,"ariaHidden":62},[61],[19,32976,32978,32981,32984,32987,32990],{"className":32977},[66],[19,32979],{"className":32980,"style":3309},[70],[19,32982,49],{"className":32983},[75,76],[19,32985],{"className":32986,"style":613},[411],[19,32988,267],{"className":32989},[335],[19,32991],{"className":32992,"style":613},[411],[19,32994,32996,32999],{"className":32995},[66],[19,32997],{"className":32998,"style":29035},[70],[19,33000,33002],{"className":33001},[75,29025],[19,33003,33005],{"className":33004},[137],[19,33006,33008],{"className":33007},[142],[19,33009,33011,33019],{"className":33010,"style":29035},[146],[19,33012,33013,33016],{"style":387},[19,33014],{"className":33015,"style":322},[154],[19,33017,49],{"className":33018},[75,76],[19,33020,33021,33024],{"style":387},[19,33022],{"className":33023,"style":322},[154],[19,33025,33027],{"className":33026,"style":29053},[29052],[19,33028,28986],{"className":33029},[75],", which contradicts the assumption. Thus, for all ",[19,33032,33034,33047],{"className":33033,"translate":23},[22],[19,33035,33037],{"className":33036},[27],[29,33038,33039],{"xmlns":31},[33,33040,33041,33045],{},[36,33042,33043],{},[39,33044,49],{},[54,33046,49],{"encoding":56},[19,33048,33050],{"className":33049,"ariaHidden":62},[61],[19,33051,33053,33056],{"className":33052},[66],[19,33054],{"className":33055,"style":3309},[70],[19,33057,49],{"className":33058},[75,76],[19,33060,33062,33095],{"className":33061,"translate":23},[22],[19,33063,33065],{"className":33064},[27],[29,33066,33067],{"xmlns":31},[33,33068,33069,33093],{},[36,33070,33071,33079,33081],{},[36,33072,33073,33075,33077],{},[43,33074,29161],{"fence":62},[39,33076,49],{},[43,33078,29161],{"fence":62},[43,33080,29448],{},[36,33082,33083,33085,33091],{},[43,33084,29161],{"fence":62},[28980,33086,33087,33089],{"accent":62},[39,33088,49],{},[43,33090,28986],{},[43,33092,29161],{"fence":62},[54,33094,29463],{"encoding":56},[19,33096,33098,33125],{"className":33097,"ariaHidden":62},[61],[19,33099,33101,33104,33116,33119,33122],{"className":33100},[66],[19,33102],{"className":33103,"style":71},[70],[19,33105,33107,33110,33113],{"className":33106},[16026],[19,33108,29161],{"className":33109,"style":16031},[81,16030],[19,33111,49],{"className":33112},[75,76],[19,33114,29161],{"className":33115,"style":16031},[88,16030],[19,33117],{"className":33118,"style":613},[411],[19,33120,29448],{"className":33121},[335],[19,33123],{"className":33124,"style":613},[411],[19,33126,33128,33131],{"className":33127},[66],[19,33129],{"className":33130,"style":71},[70],[19,33132,33134,33137,33168],{"className":33133},[16026],[19,33135,29161],{"className":33136,"style":16031},[81,16030],[19,33138,33140],{"className":33139},[75,29025],[19,33141,33143],{"className":33142},[137],[19,33144,33146],{"className":33145},[142],[19,33147,33149,33157],{"className":33148,"style":29035},[146],[19,33150,33151,33154],{"style":387},[19,33152],{"className":33153,"style":322},[154],[19,33155,49],{"className":33156},[75,76],[19,33158,33159,33162],{"style":387},[19,33160],{"className":33161,"style":322},[154],[19,33163,33165],{"className":33164,"style":29053},[29052],[19,33166,28986],{"className":33167},[75],[19,33169,29161],{"className":33170,"style":16031},[88,16030],", the power series ",[19,33173,33175,33218],{"className":33174,"translate":23},[22],[19,33176,33178],{"className":33177},[27],[29,33179,33180],{"xmlns":31},[33,33181,33182,33216],{},[36,33183,33184,33198],{},[4285,33185,33186,33188,33196],{},[43,33187,26844],{},[36,33189,33190,33192,33194],{},[39,33191,3297],{},[43,33193,267],{},[108,33195,110],{},[39,33197,26855],{"mathvariant":275},[36,33199,33200,33206],{},[103,33201,33202,33204],{},[39,33203,3484],{},[39,33205,3297],{},[269,33207,33208,33214],{},[28980,33209,33210,33212],{"accent":62},[39,33211,49],{},[43,33213,28986],{},[39,33215,3297],{},[54,33217,29595],{"encoding":56},[19,33219,33221],{"className":33220,"ariaHidden":62},[61],[19,33222,33224,33227,33290,33293],{"className":33223},[66],[19,33225],{"className":33226,"style":31929},[70],[19,33228,33230,33233],{"className":33229},[304],[19,33231,26844],{"className":33232,"style":31937},[304,27031,31936],[19,33234,33236],{"className":33235},[133],[19,33237,33239,33282],{"className":33238},[137,138],[19,33240,33242,33279],{"className":33241},[142],[19,33243,33245,33265],{"className":33244,"style":31950},[146],[19,33246,33247,33250],{"style":31953},[19,33248],{"className":33249,"style":155},[154],[19,33251,33253],{"className":33252},[159,160,161,162],[19,33254,33256,33259,33262],{"className":33255},[75,162],[19,33257,3297],{"className":33258},[75,76,162],[19,33260,267],{"className":33261},[335,162],[19,33263,110],{"className":33264},[75,162],[19,33266,33267,33270],{"style":31974},[19,33268],{"className":33269,"style":155},[154],[19,33271,33273],{"className":33272},[159,160,161,162],[19,33274,33276],{"className":33275},[75,162],[19,33277,26855],{"className":33278},[75,162],[19,33280,173],{"className":33281},[172],[19,33283,33285],{"className":33284},[142],[19,33286,33288],{"className":33287,"style":31996},[146],[19,33289],{},[19,33291],{"className":33292,"style":412},[411],[19,33294,33296,33339],{"className":33295},[75],[19,33297,33299,33302],{"className":33298},[75],[19,33300,3484],{"className":33301},[75,76],[19,33303,33305],{"className":33304},[133],[19,33306,33308,33331],{"className":33307},[137,138],[19,33309,33311,33328],{"className":33310},[142],[19,33312,33314],{"className":33313,"style":3609},[146],[19,33315,33316,33319],{"style":150},[19,33317],{"className":33318,"style":155},[154],[19,33320,33322],{"className":33321},[159,160,161,162],[19,33323,33325],{"className":33324},[75,162],[19,33326,3297],{"className":33327},[75,76,162],[19,33329,173],{"className":33330},[172],[19,33332,33334],{"className":33333},[142],[19,33335,33337],{"className":33336,"style":180},[146],[19,33338],{},[19,33340,33342,33373],{"className":33341},[75],[19,33343,33345],{"className":33344},[75,29025],[19,33346,33348],{"className":33347},[137],[19,33349,33351],{"className":33350},[142],[19,33352,33354,33362],{"className":33353,"style":29035},[146],[19,33355,33356,33359],{"style":387},[19,33357],{"className":33358,"style":322},[154],[19,33360,49],{"className":33361},[75,76],[19,33363,33364,33367],{"style":387},[19,33365],{"className":33366,"style":322},[154],[19,33368,33370],{"className":33369,"style":29053},[29052],[19,33371,28986],{"className":33372},[75],[19,33374,33376],{"className":33375},[133],[19,33377,33379],{"className":33378},[137],[19,33380,33382],{"className":33381},[142],[19,33383,33385],{"className":33384,"style":3429},[146],[19,33386,33387,33390],{"style":1643},[19,33388],{"className":33389,"style":155},[154],[19,33391,33393],{"className":33392},[159,160,161,162],[19,33394,33396],{"className":33395},[75,162],[19,33397,3297],{"className":33398},[75,76,162]," diverges.",[15,33401,33402,33403,33437,33438,33466,33467,33470,33471,33528,33529,33532,33533,3447],{},"In fact, Abel's theorem tells us about the convergence characteristics of power series, that is, the domain of convergence of a power series must be an interval centered at the origin. If we denote the length of this interval as ",[19,33404,33406,33422],{"className":33405,"translate":23},[22],[19,33407,33409],{"className":33408},[27],[29,33410,33411],{"xmlns":31},[33,33412,33413,33419],{},[36,33414,33415,33417],{},[108,33416,3518],{},[39,33418,10758],{},[54,33420,33421],{"encoding":56},"2R",[19,33423,33425],{"className":33424,"ariaHidden":62},[61],[19,33426,33428,33431,33434],{"className":33427},[66],[19,33429],{"className":33430,"style":7738},[70],[19,33432,3518],{"className":33433},[75],[19,33435,10758],{"className":33436,"style":10844},[75,76],", then ",[19,33439,33441,33454],{"className":33440,"translate":23},[22],[19,33442,33444],{"className":33443},[27],[29,33445,33446],{"xmlns":31},[33,33447,33448,33452],{},[36,33449,33450],{},[39,33451,10758],{},[54,33453,10758],{"encoding":56},[19,33455,33457],{"className":33456,"ariaHidden":62},[61],[19,33458,33460,33463],{"className":33459},[66],[19,33461],{"className":33462,"style":7738},[70],[19,33464,10758],{"className":33465,"style":10844},[75,76]," is called the ",[7600,33468,33469],{},"radius of convergence"," of the power series. We refer to ",[19,33472,33474,33498],{"className":33473,"translate":23},[22],[19,33475,33477],{"className":33476},[27],[29,33478,33479],{"xmlns":31},[33,33480,33481,33495],{},[36,33482,33483,33485,33487,33489,33491,33493],{},[43,33484,46],{"stretchy":45},[43,33486,240],{},[39,33488,10758],{},[43,33490,7830],{"separator":62},[39,33492,10758],{},[43,33494,52],{"stretchy":45},[54,33496,33497],{"encoding":56},"(-R,R)",[19,33499,33501],{"className":33500,"ariaHidden":62},[61],[19,33502,33504,33507,33510,33513,33516,33519,33522,33525],{"className":33503},[66],[19,33505],{"className":33506,"style":71},[70],[19,33508,46],{"className":33509},[81],[19,33511,240],{"className":33512},[75],[19,33514,10758],{"className":33515,"style":10844},[75,76],[19,33517,7830],{"className":33518},[8036],[19,33520],{"className":33521,"style":412},[411],[19,33523,10758],{"className":33524,"style":10844},[75,76],[19,33526,52],{"className":33527},[88]," as the ",[7600,33530,33531],{},"interval of convergence"," of the power series ",[19,33534,33536,33575],{"className":33535,"translate":23},[22],[19,33537,33539],{"className":33538},[27],[29,33540,33541],{"xmlns":31},[33,33542,33543,33573],{},[36,33544,33545,33559],{},[4285,33546,33547,33549,33557],{},[43,33548,26844],{},[36,33550,33551,33553,33555],{},[39,33552,3297],{},[43,33554,267],{},[108,33556,110],{},[39,33558,26855],{"mathvariant":275},[36,33560,33561,33567],{},[103,33562,33563,33565],{},[39,33564,3484],{},[39,33566,3297],{},[269,33568,33569,33571],{},[39,33570,49],{},[39,33572,3297],{},[54,33574,28591],{"encoding":56},[19,33576,33578],{"className":33577,"ariaHidden":62},[61],[19,33579,33581,33584,33647,33650],{"className":33580},[66],[19,33582],{"className":33583,"style":31929},[70],[19,33585,33587,33590],{"className":33586},[304],[19,33588,26844],{"className":33589,"style":31937},[304,27031,31936],[19,33591,33593],{"className":33592},[133],[19,33594,33596,33639],{"className":33595},[137,138],[19,33597,33599,33636],{"className":33598},[142],[19,33600,33602,33622],{"className":33601,"style":31950},[146],[19,33603,33604,33607],{"style":31953},[19,33605],{"className":33606,"style":155},[154],[19,33608,33610],{"className":33609},[159,160,161,162],[19,33611,33613,33616,33619],{"className":33612},[75,162],[19,33614,3297],{"className":33615},[75,76,162],[19,33617,267],{"className":33618},[335,162],[19,33620,110],{"className":33621},[75,162],[19,33623,33624,33627],{"style":31974},[19,33625],{"className":33626,"style":155},[154],[19,33628,33630],{"className":33629},[159,160,161,162],[19,33631,33633],{"className":33632},[75,162],[19,33634,26855],{"className":33635},[75,162],[19,33637,173],{"className":33638},[172],[19,33640,33642],{"className":33641},[142],[19,33643,33645],{"className":33644,"style":31996},[146],[19,33646],{},[19,33648],{"className":33649,"style":412},[411],[19,33651,33653,33696],{"className":33652},[75],[19,33654,33656,33659],{"className":33655},[75],[19,33657,3484],{"className":33658},[75,76],[19,33660,33662],{"className":33661},[133],[19,33663,33665,33688],{"className":33664},[137,138],[19,33666,33668,33685],{"className":33667},[142],[19,33669,33671],{"className":33670,"style":3609},[146],[19,33672,33673,33676],{"style":150},[19,33674],{"className":33675,"style":155},[154],[19,33677,33679],{"className":33678},[159,160,161,162],[19,33680,33682],{"className":33681},[75,162],[19,33683,3297],{"className":33684},[75,76,162],[19,33686,173],{"className":33687},[172],[19,33689,33691],{"className":33690},[142],[19,33692,33694],{"className":33693,"style":180},[146],[19,33695],{},[19,33697,33699,33702],{"className":33698},[75],[19,33700,49],{"className":33701},[75,76],[19,33703,33705],{"className":33704},[133],[19,33706,33708],{"className":33707},[137],[19,33709,33711],{"className":33710},[142],[19,33712,33714],{"className":33713,"style":3429},[146],[19,33715,33716,33719],{"style":1643},[19,33717],{"className":33718,"style":155},[154],[19,33720,33722],{"className":33721},[159,160,161,162],[19,33723,33725],{"className":33724},[75,162],[19,33726,3297],{"className":33727},[75,76,162],[15,33729,33730,33732],{},[7600,33731,9131],{}," For the power series",[19,33734,33736],{"className":33735,"translate":23},[187],[19,33737,33739,33778],{"className":33738,"translate":23},[22],[19,33740,33742],{"className":33741},[27],[29,33743,33744],{"xmlns":31,"display":196},[33,33745,33746,33776],{},[36,33747,33748,33762],{},[26840,33749,33750,33752,33760],{},[43,33751,26844],{},[36,33753,33754,33756,33758],{},[39,33755,3297],{},[43,33757,267],{},[108,33759,110],{},[39,33761,26855],{"mathvariant":275},[36,33763,33764,33770],{},[103,33765,33766,33768],{},[39,33767,3484],{},[39,33769,3297],{},[269,33771,33772,33774],{},[39,33773,49],{},[39,33775,3297],{},[54,33777,28591],{"encoding":56},[19,33779,33781],{"className":33780,"ariaHidden":62},[61],[19,33782,33784,33787,33854,33857],{"className":33783},[66],[19,33785],{"className":33786,"style":26984},[70],[19,33788,33790],{"className":33789},[304,305],[19,33791,33793,33846],{"className":33792},[137,138],[19,33794,33796,33843],{"className":33795},[142],[19,33797,33799,33819,33829],{"className":33798,"style":26997},[146],[19,33800,33801,33804],{"style":27000},[19,33802],{"className":33803,"style":27004},[154],[19,33805,33807],{"className":33806},[159,160,161,162],[19,33808,33810,33813,33816],{"className":33809},[75,162],[19,33811,3297],{"className":33812},[75,76,162],[19,33814,267],{"className":33815},[335,162],[19,33817,110],{"className":33818},[75,162],[19,33820,33821,33824],{"style":27022},[19,33822],{"className":33823,"style":27004},[154],[19,33825,33826],{},[19,33827,26844],{"className":33828},[304,27031,27032],[19,33830,33831,33834],{"style":27035},[19,33832],{"className":33833,"style":27004},[154],[19,33835,33837],{"className":33836},[159,160,161,162],[19,33838,33840],{"className":33839},[75,162],[19,33841,26855],{"className":33842},[75,162],[19,33844,173],{"className":33845},[172],[19,33847,33849],{"className":33848},[142],[19,33850,33852],{"className":33851,"style":27057},[146],[19,33853],{},[19,33855],{"className":33856,"style":412},[411],[19,33858,33860,33903],{"className":33859},[75],[19,33861,33863,33866],{"className":33862},[75],[19,33864,3484],{"className":33865},[75,76],[19,33867,33869],{"className":33868},[133],[19,33870,33872,33895],{"className":33871},[137,138],[19,33873,33875,33892],{"className":33874},[142],[19,33876,33878],{"className":33877,"style":3609},[146],[19,33879,33880,33883],{"style":150},[19,33881],{"className":33882,"style":155},[154],[19,33884,33886],{"className":33885},[159,160,161,162],[19,33887,33889],{"className":33888},[75,162],[19,33890,3297],{"className":33891},[75,76,162],[19,33893,173],{"className":33894},[172],[19,33896,33898],{"className":33897},[142],[19,33899,33901],{"className":33900,"style":180},[146],[19,33902],{},[19,33904,33906,33909],{"className":33905},[75],[19,33907,49],{"className":33908},[75,76],[19,33910,33912],{"className":33911},[133],[19,33913,33915],{"className":33914},[137],[19,33916,33918],{"className":33917},[142],[19,33919,33921],{"className":33920,"style":4148},[146],[19,33922,33923,33926],{"style":648},[19,33924],{"className":33925,"style":155},[154],[19,33927,33929],{"className":33928},[159,160,161,162],[19,33930,33932],{"className":33931},[75,162],[19,33933,3297],{"className":33934},[75,76,162],[15,33936,33937,33938,34215],{},"If ",[19,33939,33941,33991],{"className":33940,"translate":23},[22],[19,33942,33944],{"className":33943},[27],[29,33945,33946],{"xmlns":31},[33,33947,33948,33988],{},[36,33949,33950,33966,33983,33985],{},[103,33951,33952,33958],{},[36,33953,33954,33956],{},[39,33955,208],{},[43,33957,211],{},[36,33959,33960,33962,33964],{},[39,33961,3297],{},[43,33963,218],{},[39,33965,26855],{"mathvariant":275},[33967,33968,33969,33981],"mroot",{},[36,33970,33971,33973,33979],{},[43,33972,29161],{"fence":62},[103,33974,33975,33977],{},[39,33976,3484],{},[39,33978,3297],{},[43,33980,29161],{"fence":62},[39,33982,3297],{},[43,33984,267],{},[39,33986,33987],{},"ρ",[54,33989,33990],{"encoding":56},"\\lim_{n \\to \\infty}{\\sqrt[n]{\\left| a_{n} \\right|}} = \\rho",[19,33992,33994,34205],{"className":33993,"ariaHidden":62},[61],[19,33995,33997,34001,34051,34054,34196,34199,34202],{"className":33996},[66],[19,33998],{"className":33999,"style":34000},[70],"height:1.24em;vertical-align:-0.305em;",[19,34002,34004,34007],{"className":34003},[304],[19,34005,208],{"className":34006},[304],[19,34008,34010],{"className":34009},[133],[19,34011,34013,34043],{"className":34012},[137,138],[19,34014,34016,34040],{"className":34015},[142],[19,34017,34019],{"className":34018,"style":3609},[146],[19,34020,34022,34025],{"style":34021},"top:-2.55em;margin-right:0.05em;",[19,34023],{"className":34024,"style":155},[154],[19,34026,34028],{"className":34027},[159,160,161,162],[19,34029,34031,34034,34037],{"className":34030},[75,162],[19,34032,3297],{"className":34033},[75,76,162],[19,34035,218],{"className":34036},[335,162],[19,34038,26855],{"className":34039},[75,162],[19,34041,173],{"className":34042},[172],[19,34044,34046],{"className":34045},[142],[19,34047,34049],{"className":34048,"style":180},[146],[19,34050],{},[19,34052],{"className":34053,"style":412},[411],[19,34055,34057],{"className":34056},[75],[19,34058,34061,34090],{"className":34059},[75,34060],"sqrt",[19,34062,34065],{"className":34063},[34064],"root",[19,34066,34068],{"className":34067},[137],[19,34069,34071],{"className":34070},[142],[19,34072,34075],{"className":34073,"style":34074},[146],"height:0.5933em;",[19,34076,34078,34081],{"style":34077},"top:-2.878em;",[19,34079],{"className":34080,"style":361},[154],[19,34082,34084],{"className":34083},[159,160,366,162],[19,34085,34087],{"className":34086},[75,162],[19,34088,3297],{"className":34089},[75,76,162],[19,34091,34093,34187],{"className":34092},[137,138],[19,34094,34096,34184],{"className":34095},[142],[19,34097,34100,34164],{"className":34098,"style":34099},[146],"height:0.935em;",[19,34101,34105,34108],{"className":34102,"style":34104},[34103],"svg-align","top:-3.2em;",[19,34106],{"className":34107,"style":30456},[154],[19,34109,34112],{"className":34110,"style":34111},[75],"padding-left:1em;",[19,34113,34115,34118,34161],{"className":34114},[16026],[19,34116,29161],{"className":34117,"style":16031},[81,16030],[19,34119,34121,34124],{"className":34120},[75],[19,34122,3484],{"className":34123},[75,76],[19,34125,34127],{"className":34126},[133],[19,34128,34130,34153],{"className":34129},[137,138],[19,34131,34133,34150],{"className":34132},[142],[19,34134,34136],{"className":34135,"style":3609},[146],[19,34137,34138,34141],{"style":150},[19,34139],{"className":34140,"style":155},[154],[19,34142,34144],{"className":34143},[159,160,161,162],[19,34145,34147],{"className":34146},[75,162],[19,34148,3297],{"className":34149},[75,76,162],[19,34151,173],{"className":34152},[172],[19,34154,34156],{"className":34155},[142],[19,34157,34159],{"className":34158,"style":180},[146],[19,34160],{},[19,34162,29161],{"className":34163,"style":16031},[88,16030],[19,34165,34167,34170],{"style":34166},"top:-2.895em;",[19,34168],{"className":34169,"style":30456},[154],[19,34171,34175],{"className":34172,"style":34174},[34173],"hide-tail","min-width:1.02em;height:1.28em;",[30461,34176,34181],{"xmlns":30463,"width":34177,"height":34178,"viewBox":34179,"preserveAspectRatio":34180},"400em","1.28em","0 0 400000 1296","xMinYMin slice",[30468,34182],{"d":34183},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[19,34185,173],{"className":34186},[172],[19,34188,34190],{"className":34189},[142],[19,34191,34194],{"className":34192,"style":34193},[146],"height:0.305em;",[19,34195],{},[19,34197],{"className":34198,"style":613},[411],[19,34200,267],{"className":34201},[335],[19,34203],{"className":34204,"style":613},[411],[19,34206,34208,34212],{"className":34207},[66],[19,34209],{"className":34210,"style":34211},[70],"height:0.625em;vertical-align:-0.1944em;",[19,34213,33987],{"className":34214},[75,76],", then 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\\to \\infty}{\\sqrt[n]{\\left| a_{n}x^{n} \\right|}} = \\lim_{n \\to \\infty}{\\sqrt[n]{\\left| a_{n} \\right|}\\left| x \\right|} = \\rho\\left| x \\right|",[19,35106,35108,35343,35554],{"className":35107,"ariaHidden":62},[61],[19,35109,35111,35115,35169,35172,35334,35337,35340],{"className":35110},[66],[19,35112],{"className":35113,"style":35114},[70],"height:1.6839em;vertical-align:-0.7em;",[19,35116,35118],{"className":35117},[304,305],[19,35119,35121,35160],{"className":35120},[137,138],[19,35122,35124,35157],{"className":35123},[142],[19,35125,35127,35147],{"className":35126,"style":315},[146],[19,35128,35129,35132],{"style":318},[19,35130],{"className":35131,"style":322},[154],[19,35133,35135],{"className":35134},[159,160,161,162],[19,35136,35138,35141,35144],{"className":35137},[75,162],[19,35139,3297],{"className":35140},[75,76,162],[19,35142,218],{"className":35143},[335,162],[19,35145,26855],{"className":35146},[75,162],[19,35148,35149,35152],{"style":387},[19,35150],{"className":35151,"style":322},[154],[19,35153,35154],{},[19,35155,208],{"className":35156},[304],[19,35158,173],{"className":35159},[172],[19,35161,35163],{"className":35162},[142],[19,35164,35167],{"className":35165,"style":35166},[146],"height:0.7em;",[19,35168],{},[19,35170],{"className":35171,"style":412},[411],[19,35173,35175],{"className":35174},[75],[19,35176,35178,35206],{"className":35177},[75,34060],[19,35179,35181],{"className":35180},[34064],[19,35182,35184],{"className":35183},[137],[19,35185,35187],{"className":35186},[142],[19,35188,35191],{"className":35189,"style":35190},[146],"height:0.6519em;",[19,35192,35194,35197],{"style":35193},"top:-2.9367em;",[19,35195],{"className":35196,"style":361},[154],[19,35198,35200],{"className":35199},[159,160,366,162],[19,35201,35203],{"className":35202},[75,162],[19,35204,3297],{"className":35205},[75,76,162],[19,35207,35209,35325],{"className":35208},[137,138],[19,35210,35212,35322],{"className":35211},[142],[19,35213,35216,35309],{"className":35214,"style":35215},[146],"height:0.9839em;",[19,35217,35219,35222],{"className":35218,"style":34104},[34103],[19,35220],{"className":35221,"style":30456},[154],[19,35223,35225],{"className":35224,"style":34111},[75],[19,35226,35228,35231,35274,35306],{"className":35227},[16026],[19,35229,29161],{"className":35230,"style":16031},[81,16030],[19,35232,35234,35237],{"className":35233},[75],[19,35235,3484],{"className":35236},[75,76],[19,35238,35240],{"className":35239},[133],[19,35241,35243,35266],{"className":35242},[137,138],[19,35244,35246,35263],{"className":35245},[142],[19,35247,35249],{"className":35248,"style":3609},[146],[19,35250,35251,35254],{"style":150},[19,35252],{"className":35253,"style":155},[154],[19,35255,35257],{"className":35256},[159,160,161,162],[19,35258,35260],{"className":35259},[75,162],[19,35261,3297],{"className":35262},[75,76,162],[19,35264,173],{"className":35265},[172],[19,35267,35269],{"className":35268},[142],[19,35270,35272],{"className":35271,"style":180},[146],[19,35273],{},[19,35275,35277,35280],{"className":35276},[75],[19,35278,49],{"className":35279},[75,76],[19,35281,35283],{"className":35282},[133],[19,35284,35286],{"className":35285},[137],[19,35287,35289],{"className":35288},[142],[19,35290,35292],{"className":35291,"style":31117},[146],[19,35293,35294,35297],{"style":14745},[19,35295],{"className":35296,"style":155},[154],[19,35298,35300],{"className":35299},[159,160,161,162],[19,35301,35303],{"className":35302},[75,162],[19,35304,3297],{"className":35305},[75,76,162],[19,35307,29161],{"className":35308,"style":16031},[88,16030],[19,35310,35312,35315],{"style":35311},"top:-2.9439em;",[19,35313],{"className":35314,"style":30456},[154],[19,35316,35318],{"className":35317,"style":34174},[34173],[30461,35319,35320],{"xmlns":30463,"width":34177,"height":34178,"viewBox":34179,"preserveAspectRatio":34180},[30468,35321],{"d":34183},[19,35323,173],{"className":35324},[172],[19,35326,35328],{"className":35327},[142],[19,35329,35332],{"className":35330,"style":35331},[146],"height:0.2561em;",[19,35333],{},[19,35335],{"className":35336,"style":613},[411],[19,35338,267],{"className":35339},[335],[19,35341],{"className":35342,"style":613},[411],[19,35344,35346,35349,35402,35405,35545,35548,35551],{"className":35345},[66],[19,35347],{"className":35348,"style":35114},[70],[19,35350,35352],{"className":35351},[304,305],[19,35353,35355,35394],{"className":35354},[137,138],[19,35356,35358,35391],{"className":35357},[142],[19,35359,35361,35381],{"className":35360,"style":315},[146],[19,35362,35363,35366],{"style":318},[19,35364],{"className":35365,"style":322},[154],[19,35367,35369],{"className":35368},[159,160,161,162],[19,35370,35372,35375,35378],{"className":35371},[75,162],[19,35373,3297],{"className":35374},[75,76,162],[19,35376,218],{"className":35377},[335,162],[19,35379,26855],{"className":35380},[75,162],[19,35382,35383,35386],{"style":387},[19,35384],{"className":35385,"style":322},[154],[19,35387,35388],{},[19,35389,208],{"className":35390},[304],[19,35392,173],{"className":35393},[172],[19,35395,35397],{"className":35396},[142],[19,35398,35400],{"className":35399,"style":35166},[146],[19,35401],{},[19,35403],{"className":35404,"style":412},[411],[19,35406,35408,35530,35533],{"className":35407},[75],[19,35409,35411,35437],{"className":35410},[75,34060],[19,35412,35414],{"className":35413},[34064],[19,35415,35417],{"className":35416},[137],[19,35418,35420],{"className":35419},[142],[19,35421,35423],{"className":35422,"style":35190},[146],[19,35424,35425,35428],{"style":35193},[19,35426],{"className":35427,"style":361},[154],[19,35429,35431],{"className":35430},[159,160,366,162],[19,35432,35434],{"className":35433},[75,162],[19,35435,3297],{"className":35436},[75,76,162],[19,35438,35440,35522],{"className":35439},[137,138],[19,35441,35443,35519],{"className":35442},[142],[19,35444,35446,35507],{"className":35445,"style":35215},[146],[19,35447,35449,35452],{"className":35448,"style":34104},[34103],[19,35450],{"className":35451,"style":30456},[154],[19,35453,35455],{"className":35454,"style":34111},[75],[19,35456,35458,35461,35504],{"className":35457},[16026],[19,35459,29161],{"className":35460,"style":16031},[81,16030],[19,35462,35464,35467],{"className":35463},[75],[19,35465,3484],{"className":35466},[75,76],[19,35468,35470],{"className":35469},[133],[19,35471,35473,35496],{"className":35472},[137,138],[19,35474,35476,35493],{"className":35475},[142],[19,35477,35479],{"className":35478,"style":3609},[146],[19,35480,35481,35484],{"style":150},[19,35482],{"className":35483,"style":155},[154],[19,35485,35487],{"className":35486},[159,160,161,162],[19,35488,35490],{"className":35489},[75,162],[19,35491,3297],{"className":35492},[75,76,162],[19,35494,173],{"className":35495},[172],[19,35497,35499],{"className":35498},[142],[19,35500,35502],{"className":35501,"style":180},[146],[19,35503],{},[19,35505,29161],{"className":35506,"style":16031},[88,16030],[19,35508,35509,35512],{"style":35311},[19,35510],{"className":35511,"style":30456},[154],[19,35513,35515],{"className":35514,"style":34174},[34173],[30461,35516,35517],{"xmlns":30463,"width":34177,"height":34178,"viewBox":34179,"preserveAspectRatio":34180},[30468,35518],{"d":34183},[19,35520,173],{"className":35521},[172],[19,35523,35525],{"className":35524},[142],[19,35526,35528],{"className":35527,"style":35331},[146],[19,35529],{},[19,35531],{"className":35532,"style":412},[411],[19,35534,35536,35539,35542],{"className":35535},[16026],[19,35537,29161],{"className":35538,"style":16031},[81,16030],[19,35540,49],{"className":35541},[75,76],[19,35543,29161],{"className":35544,"style":16031},[88,16030],[19,35546],{"className":35547,"style":613},[411],[19,35549,267],{"className":35550},[335],[19,35552],{"className":35553,"style":613},[411],[19,35555,35557,35560,35563,35566],{"className":35556},[66],[19,35558],{"className":35559,"style":71},[70],[19,35561,33987],{"className":35562},[75,76],[19,35564],{"className":35565,"style":412},[411],[19,35567,35569,35572,35575],{"className":35568},[16026],[19,35570,29161],{"className":35571,"style":16031},[81,16030],[19,35573,49],{"className":35574},[75,76],[19,35576,29161],{"className":35577,"style":16031},[88,16030],[15,35579,35580,35581,13307,35655,35863,35864,35938,35939,36016,36017,36090,36091,36210,36211,13307,36261,36334,36335,36363,36364,36210,36419,13307,36475,36334,36548,36576,36577,36363,36628,3447],{},"According to the root test for positive series, when ",[19,35582,35584,35610],{"className":35583,"translate":23},[22],[19,35585,35587],{"className":35586},[27],[29,35588,35589],{"xmlns":31},[33,35590,35591,35607],{},[36,35592,35593,35595,35603,35605],{},[39,35594,33987],{},[36,35596,35597,35599,35601],{},[43,35598,29161],{"fence":62},[39,35600,49],{},[43,35602,29161],{"fence":62},[43,35604,29168],{},[108,35606,3495],{},[54,35608,35609],{"encoding":56},"\\rho\\left| x \\right| \u003C 1",[19,35611,35613,35646],{"className":35612,"ariaHidden":62},[61],[19,35614,35616,35619,35622,35625,35637,35640,35643],{"className":35615},[66],[19,35617],{"className":35618,"style":71},[70],[19,35620,33987],{"className":35621},[75,76],[19,35623],{"className":35624,"style":412},[411],[19,35626,35628,35631,35634],{"className":35627},[16026],[19,35629,29161],{"className":35630,"style":16031},[81,16030],[19,35632,49],{"className":35633},[75,76],[19,35635,29161],{"className":35636,"style":16031},[88,16030],[19,35638],{"className":35639,"style":613},[411],[19,35641,29168],{"className":35642},[335],[19,35644],{"className":35645,"style":613},[411],[19,35647,35649,35652],{"className":35648},[66],[19,35650],{"className":35651,"style":1214},[70],[19,35653,3495],{"className":35654},[75],[19,35656,35658,35701],{"className":35657,"translate":23},[22],[19,35659,35661],{"className":35660},[27],[29,35662,35663],{"xmlns":31},[33,35664,35665,35699],{},[36,35666,35667,35681],{},[4285,35668,35669,35671,35679],{},[43,35670,26844],{},[36,35672,35673,35675,35677],{},[39,35674,3297],{},[43,35676,267],{},[108,35678,110],{},[39,35680,26855],{"mathvariant":275},[36,35682,35683,35685,35691,35697],{},[43,35684,29161],{"fence":62},[103,35686,35687,35689],{},[39,35688,3484],{},[39,35690,3297],{},[269,35692,35693,35695],{},[39,35694,49],{},[39,35696,3297],{},[43,35698,29161],{"fence":62},[54,35700,34824],{"encoding":56},[19,35702,35704],{"className":35703,"ariaHidden":62},[61],[19,35705,35707,35710,35773,35776],{"className":35706},[66],[19,35708],{"className":35709,"style":31929},[70],[19,35711,35713,35716],{"className":35712},[304],[19,35714,26844],{"className":35715,"style":31937},[304,27031,31936],[19,35717,35719],{"className":35718},[133],[19,35720,35722,35765],{"className":35721},[137,138],[19,35723,35725,35762],{"className":35724},[142],[19,35726,35728,35748],{"className":35727,"style":31950},[146],[19,35729,35730,35733],{"style":31953},[19,35731],{"className":35732,"style":155},[154],[19,35734,35736],{"className":35735},[159,160,161,162],[19,35737,35739,35742,35745],{"className":35738},[75,162],[19,35740,3297],{"className":35741},[75,76,162],[19,35743,267],{"className":35744},[335,162],[19,35746,110],{"className":35747},[75,162],[19,35749,35750,35753],{"style":31974},[19,35751],{"className":35752,"style":155},[154],[19,35754,35756],{"className":35755},[159,160,161,162],[19,35757,35759],{"className":35758},[75,162],[19,35760,26855],{"className":35761},[75,162],[19,35763,173],{"className":35764},[172],[19,35766,35768],{"className":35767},[142],[19,35769,35771],{"className":35770,"style":31996},[146],[19,35772],{},[19,35774],{"className":35775,"style":412},[411],[19,35777,35779],{"className":35778},[75],[19,35780,35782,35785,35828,35860],{"className":35781},[16026],[19,35783,29161],{"className":35784,"style":16031},[81,16030],[19,35786,35788,35791],{"className":35787},[75],[19,35789,3484],{"className":35790},[75,76],[19,35792,35794],{"className":35793},[133],[19,35795,35797,35820],{"className":35796},[137,138],[19,35798,35800,35817],{"className":35799},[142],[19,35801,35803],{"className":35802,"style":3609},[146],[19,35804,35805,35808],{"style":150},[19,35806],{"className":35807,"style":155},[154],[19,35809,35811],{"className":35810},[159,160,161,162],[19,35812,35814],{"className":35813},[75,162],[19,35815,3297],{"className":35816},[75,76,162],[19,35818,173],{"className":35819},[172],[19,35821,35823],{"className":35822},[142],[19,35824,35826],{"className":35825,"style":180},[146],[19,35827],{},[19,35829,35831,35834],{"className":35830},[75],[19,35832,49],{"className":35833},[75,76],[19,35835,35837],{"className":35836},[133],[19,35838,35840],{"className":35839},[137],[19,35841,35843],{"className":35842},[142],[19,35844,35846],{"className":35845,"style":3429},[146],[19,35847,35848,35851],{"style":1643},[19,35849],{"className":35850,"style":155},[154],[19,35852,35854],{"className":35853},[159,160,161,162],[19,35855,35857],{"className":35856},[75,162],[19,35858,3297],{"className":35859},[75,76,162],[19,35861,29161],{"className":35862,"style":16031},[88,16030]," converges; when ",[19,35865,35867,35893],{"className":35866,"translate":23},[22],[19,35868,35870],{"className":35869},[27],[29,35871,35872],{"xmlns":31},[33,35873,35874,35890],{},[36,35875,35876,35878,35886,35888],{},[39,35877,33987],{},[36,35879,35880,35882,35884],{},[43,35881,29161],{"fence":62},[39,35883,49],{},[43,35885,29161],{"fence":62},[43,35887,29448],{},[108,35889,3495],{},[54,35891,35892],{"encoding":56},"\\rho\\left| x \\right| > 1",[19,35894,35896,35929],{"className":35895,"ariaHidden":62},[61],[19,35897,35899,35902,35905,35908,35920,35923,35926],{"className":35898},[66],[19,35900],{"className":35901,"style":71},[70],[19,35903,33987],{"className":35904},[75,76],[19,35906],{"className":35907,"style":412},[411],[19,35909,35911,35914,35917],{"className":35910},[16026],[19,35912,29161],{"className":35913,"style":16031},[81,16030],[19,35915,49],{"className":35916},[75,76],[19,35918,29161],{"className":35919,"style":16031},[88,16030],[19,35921],{"className":35922,"style":613},[411],[19,35924,29448],{"className":35925},[335],[19,35927],{"className":35928,"style":613},[411],[19,35930,35932,35935],{"className":35931},[66],[19,35933],{"className":35934,"style":1214},[70],[19,35936,3495],{"className":35937},[75]," the power series diverges. Therefore, when ",[19,35940,35942,35965],{"className":35941,"translate":23},[22],[19,35943,35945],{"className":35944},[27],[29,35946,35947],{"xmlns":31},[33,35948,35949,35963],{},[36,35950,35951,35953,35955,35957,35959,35961],{},[108,35952,110],{},[43,35954,29168],{},[39,35956,33987],{},[43,35958,29168],{},[43,35960,2351],{},[39,35962,26855],{"mathvariant":275},[54,35964,34289],{"encoding":56},[19,35966,35968,35986,36004],{"className":35967,"ariaHidden":62},[61],[19,35969,35971,35974,35977,35980,35983],{"className":35970},[66],[19,35972],{"className":35973,"style":34299},[70],[19,35975,110],{"className":35976},[75],[19,35978],{"className":35979,"style":613},[411],[19,35981,29168],{"className":35982},[335],[19,35984],{"className":35985,"style":613},[411],[19,35987,35989,35992,35995,35998,36001],{"className":35988},[66],[19,35990],{"className":35991,"style":17418},[70],[19,35993,33987],{"className":35994},[75,76],[19,35996],{"className":35997,"style":613},[411],[19,35999,29168],{"className":36000},[335],[19,36002],{"className":36003,"style":613},[411],[19,36005,36007,36010,36013],{"className":36006},[66],[19,36008],{"className":36009,"style":4003},[70],[19,36011,2351],{"className":36012},[75],[19,36014,26855],{"className":36015},[75],", from ",[19,36018,36020,36045],{"className":36019,"translate":23},[22],[19,36021,36023],{"className":36022},[27],[29,36024,36025],{"xmlns":31},[33,36026,36027,36043],{},[36,36028,36029,36031,36039,36041],{},[39,36030,33987],{},[36,36032,36033,36035,36037],{},[43,36034,29161],{"fence":62},[39,36036,49],{},[43,36038,29161],{"fence":62},[43,36040,29168],{},[108,36042,3495],{},[54,36044,35609],{"encoding":56},[19,36046,36048,36081],{"className":36047,"ariaHidden":62},[61],[19,36049,36051,36054,36057,36060,36072,36075,36078],{"className":36050},[66],[19,36052],{"className":36053,"style":71},[70],[19,36055,33987],{"className":36056},[75,76],[19,36058],{"className":36059,"style":412},[411],[19,36061,36063,36066,36069],{"className":36062},[16026],[19,36064,29161],{"className":36065,"style":16031},[81,16030],[19,36067,49],{"className":36068},[75,76],[19,36070,29161],{"className":36071,"style":16031},[88,16030],[19,36073],{"className":36074,"style":613},[411],[19,36076,29168],{"className":36077},[335],[19,36079],{"className":36080,"style":613},[411],[19,36082,36084,36087],{"className":36083},[66],[19,36085],{"className":36086,"style":1214},[70],[19,36088,3495],{"className":36089},[75],", we obtain the radius of convergence ",[19,36092,36094,36115],{"className":36093,"translate":23},[22],[19,36095,36097],{"className":36096},[27],[29,36098,36099],{"xmlns":31},[33,36100,36101,36113],{},[36,36102,36103,36105,36107],{},[39,36104,10758],{},[43,36106,267],{},[226,36108,36109,36111],{},[108,36110,3495],{},[39,36112,33987],{},[54,36114,34367],{"encoding":56},[19,36116,36118,36136],{"className":36117,"ariaHidden":62},[61],[19,36119,36121,36124,36127,36130,36133],{"className":36120},[66],[19,36122],{"className":36123,"style":7738},[70],[19,36125,10758],{"className":36126,"style":10844},[75,76],[19,36128],{"className":36129,"style":613},[411],[19,36131,267],{"className":36132},[335],[19,36134],{"className":36135,"style":613},[411],[19,36137,36139,36142],{"className":36138},[66],[19,36140],{"className":36141,"style":34395},[70],[19,36143,36145,36148,36207],{"className":36144},[75],[19,36146],{"className":36147},[81,422],[19,36149,36151],{"className":36150},[226],[19,36152,36154,36199],{"className":36153},[137,138],[19,36155,36157,36196],{"className":36156},[142],[19,36158,36160,36174,36182],{"className":36159,"style":34414},[146],[19,36161,36162,36165],{"style":30504},[19,36163],{"className":36164,"style":322},[154],[19,36166,36168],{"className":36167},[159,160,161,162],[19,36169,36171],{"className":36170},[75,162],[19,36172,33987],{"className":36173},[75,76,162],[19,36175,36176,36179],{"style":504},[19,36177],{"className":36178,"style":322},[154],[19,36180],{"className":36181,"style":512},[511],[19,36183,36184,36187],{"style":30556},[19,36185],{"className":36186,"style":322},[154],[19,36188,36190],{"className":36189},[159,160,161,162],[19,36191,36193],{"className":36192},[75,162],[19,36194,3495],{"className":36195},[75,162],[19,36197,173],{"className":36198},[172],[19,36200,36202],{"className":36201},[142],[19,36203,36205],{"className":36204,"style":34460},[146],[19,36206],{},[19,36208],{"className":36209},[88,422],". When ",[19,36212,36214,36231],{"className":36213,"translate":23},[22],[19,36215,36217],{"className":36216},[27],[29,36218,36219],{"xmlns":31},[33,36220,36221,36229],{},[36,36222,36223,36225,36227],{},[39,36224,33987],{},[43,36226,267],{},[108,36228,110],{},[54,36230,34531],{"encoding":56},[19,36232,36234,36252],{"className":36233,"ariaHidden":62},[61],[19,36235,36237,36240,36243,36246,36249],{"className":36236},[66],[19,36238],{"className":36239,"style":34211},[70],[19,36241,33987],{"className":36242},[75,76],[19,36244],{"className":36245,"style":613},[411],[19,36247,267],{"className":36248},[335],[19,36250],{"className":36251,"style":613},[411],[19,36253,36255,36258],{"className":36254},[66],[19,36256],{"className":36257,"style":1214},[70],[19,36259,110],{"className":36260},[75],[19,36262,36264,36289],{"className":36263,"translate":23},[22],[19,36265,36267],{"className":36266},[27],[29,36268,36269],{"xmlns":31},[33,36270,36271,36287],{},[36,36272,36273,36275,36283,36285],{},[39,36274,33987],{},[36,36276,36277,36279,36281],{},[43,36278,29161],{"fence":62},[39,36280,49],{},[43,36282,29161],{"fence":62},[43,36284,29168],{},[108,36286,3495],{},[54,36288,35609],{"encoding":56},[19,36290,36292,36325],{"className":36291,"ariaHidden":62},[61],[19,36293,36295,36298,36301,36304,36316,36319,36322],{"className":36294},[66],[19,36296],{"className":36297,"style":71},[70],[19,36299,33987],{"className":36300},[75,76],[19,36302],{"className":36303,"style":412},[411],[19,36305,36307,36310,36313],{"className":36306},[16026],[19,36308,29161],{"className":36309,"style":16031},[81,16030],[19,36311,49],{"className":36312},[75,76],[19,36314,29161],{"className":36315,"style":16031},[88,16030],[19,36317],{"className":36318,"style":613},[411],[19,36320,29168],{"className":36321},[335],[19,36323],{"className":36324,"style":613},[411],[19,36326,36328,36331],{"className":36327},[66],[19,36329],{"className":36330,"style":1214},[70],[19,36332,3495],{"className":36333},[75]," holds for any ",[19,36336,36338,36351],{"className":36337,"translate":23},[22],[19,36339,36341],{"className":36340},[27],[29,36342,36343],{"xmlns":31},[33,36344,36345,36349],{},[36,36346,36347],{},[39,36348,49],{},[54,36350,49],{"encoding":56},[19,36352,36354],{"className":36353,"ariaHidden":62},[61],[19,36355,36357,36360],{"className":36356},[66],[19,36358],{"className":36359,"style":3309},[70],[19,36361,49],{"className":36362},[75,76],", so ",[19,36365,36367,36386],{"className":36366,"translate":23},[22],[19,36368,36370],{"className":36369},[27],[29,36371,36372],{"xmlns":31},[33,36373,36374,36384],{},[36,36375,36376,36378,36380,36382],{},[39,36377,10758],{},[43,36379,267],{},[43,36381,2351],{},[39,36383,26855],{"mathvariant":275},[54,36385,34584],{"encoding":56},[19,36387,36389,36407],{"className":36388,"ariaHidden":62},[61],[19,36390,36392,36395,36398,36401,36404],{"className":36391},[66],[19,36393],{"className":36394,"style":7738},[70],[19,36396,10758],{"className":36397,"style":10844},[75,76],[19,36399],{"className":36400,"style":613},[411],[19,36402,267],{"className":36403},[335],[19,36405],{"className":36406,"style":613},[411],[19,36408,36410,36413,36416],{"className":36409},[66],[19,36411],{"className":36412,"style":4003},[70],[19,36414,2351],{"className":36415},[75],[19,36417,26855],{"className":36418},[75],[19,36420,36422,36442],{"className":36421,"translate":23},[22],[19,36423,36425],{"className":36424},[27],[29,36426,36427],{"xmlns":31},[33,36428,36429,36439],{},[36,36430,36431,36433,36435,36437],{},[39,36432,33987],{},[43,36434,267],{},[43,36436,2351],{},[39,36438,26855],{"mathvariant":275},[54,36440,36441],{"encoding":56},"\\rho = +\\infty",[19,36443,36445,36463],{"className":36444,"ariaHidden":62},[61],[19,36446,36448,36451,36454,36457,36460],{"className":36447},[66],[19,36449],{"className":36450,"style":34211},[70],[19,36452,33987],{"className":36453},[75,76],[19,36455],{"className":36456,"style":613},[411],[19,36458,267],{"className":36459},[335],[19,36461],{"className":36462,"style":613},[411],[19,36464,36466,36469,36472],{"className":36465},[66],[19,36467],{"className":36468,"style":4003},[70],[19,36470,2351],{"className":36471},[75],[19,36473,26855],{"className":36474},[75],[19,36476,36478,36503],{"className":36477,"translate":23},[22],[19,36479,36481],{"className":36480},[27],[29,36482,36483],{"xmlns":31},[33,36484,36485,36501],{},[36,36486,36487,36489,36497,36499],{},[39,36488,33987],{},[36,36490,36491,36493,36495],{},[43,36492,29161],{"fence":62},[39,36494,49],{},[43,36496,29161],{"fence":62},[43,36498,29448],{},[108,36500,3495],{},[54,36502,35892],{"encoding":56},[19,36504,36506,36539],{"className":36505,"ariaHidden":62},[61],[19,36507,36509,36512,36515,36518,36530,36533,36536],{"className":36508},[66],[19,36510],{"className":36511,"style":71},[70],[19,36513,33987],{"className":36514},[75,76],[19,36516],{"className":36517,"style":412},[411],[19,36519,36521,36524,36527],{"className":36520},[16026],[19,36522,29161],{"className":36523,"style":16031},[81,16030],[19,36525,49],{"className":36526},[75,76],[19,36528,29161],{"className":36529,"style":16031},[88,16030],[19,36531],{"className":36532,"style":613},[411],[19,36534,29448],{"className":36535},[335],[19,36537],{"className":36538,"style":613},[411],[19,36540,36542,36545],{"className":36541},[66],[19,36543],{"className":36544,"style":1214},[70],[19,36546,3495],{"className":36547},[75],[19,36549,36551,36564],{"className":36550,"translate":23},[22],[19,36552,36554],{"className":36553},[27],[29,36555,36556],{"xmlns":31},[33,36557,36558,36562],{},[36,36559,36560],{},[39,36561,49],{},[54,36563,49],{"encoding":56},[19,36565,36567],{"className":36566,"ariaHidden":62},[61],[19,36568,36570,36573],{"className":36569},[66],[19,36571],{"className":36572,"style":3309},[70],[19,36574,49],{"className":36575},[75,76]," except ",[19,36578,36580,36598],{"className":36579,"translate":23},[22],[19,36581,36583],{"className":36582},[27],[29,36584,36585],{"xmlns":31},[33,36586,36587,36595],{},[36,36588,36589,36591,36593],{},[39,36590,49],{},[43,36592,267],{},[108,36594,110],{},[54,36596,36597],{"encoding":56},"x = 0",[19,36599,36601,36619],{"className":36600,"ariaHidden":62},[61],[19,36602,36604,36607,36610,36613,36616],{"className":36603},[66],[19,36605],{"className":36606,"style":3309},[70],[19,36608,49],{"className":36609},[75,76],[19,36611],{"className":36612,"style":613},[411],[19,36614,267],{"className":36615},[335],[19,36617],{"className":36618,"style":613},[411],[19,36620,36622,36625],{"className":36621},[66],[19,36623],{"className":36624,"style":1214},[70],[19,36626,110],{"className":36627},[75],[19,36629,36631,36648],{"className":36630,"translate":23},[22],[19,36632,36634],{"className":36633},[27],[29,36635,36636],{"xmlns":31},[33,36637,36638,36646],{},[36,36639,36640,36642,36644],{},[39,36641,10758],{},[43,36643,267],{},[108,36645,110],{},[54,36647,34740],{"encoding":56},[19,36649,36651,36669],{"className":36650,"ariaHidden":62},[61],[19,36652,36654,36657,36660,36663,36666],{"className":36653},[66],[19,36655],{"className":36656,"style":7738},[70],[19,36658,10758],{"className":36659,"style":10844},[75,76],[19,36661],{"className":36662,"style":613},[411],[19,36664,267],{"className":36665},[335],[19,36667],{"className":36668,"style":613},[411],[19,36670,36672,36675],{"className":36671},[66],[19,36673],{"className":36674,"style":1214},[70],[19,36676,110],{"className":36677},[75],[15,36679,36680,36682],{},[7600,36681,16911],{}," If",[19,36684,36686],{"className":36685,"translate":23},[187],[19,36687,36689,36753],{"className":36688,"translate":23},[22],[19,36690,36692],{"className":36691},[27],[29,36693,36694],{"xmlns":31,"display":196},[33,36695,36696,36750],{},[36,36697,36698,36714,36746,36748],{},[202,36699,36700,36706],{},[36,36701,36702,36704],{},[39,36703,208],{},[43,36705,211],{},[36,36707,36708,36710,36712],{},[39,36709,3297],{},[43,36711,218],{},[39,36713,26855],{"mathvariant":275},[226,36715,36716,36734],{},[36,36717,36718,36720,36732],{},[43,36719,29161],{"fence":62},[103,36721,36722,36724],{},[39,36723,3484],{},[36,36725,36726,36728,36730],{},[39,36727,3297],{},[43,36729,2351],{},[108,36731,3495],{},[43,36733,29161],{"fence":62},[36,36735,36736,36738,36744],{},[43,36737,29161],{"fence":62},[103,36739,36740,36742],{},[39,36741,3484],{},[39,36743,3297],{},[43,36745,29161],{"fence":62},[43,36747,267],{},[39,36749,33987],{},[54,36751,36752],{"encoding":56},"\\lim_{n \\to \\infty}{\\frac{\\left| a_{n+1} \\right|}{\\left| a_{n} \\right|}} = \\rho",[19,36754,36756,36997],{"className":36755,"ariaHidden":62},[61],[19,36757,36759,36762,36815,36818,36988,36991,36994],{"className":36758},[66],[19,36760],{"className":36761,"style":11242},[70],[19,36763,36765],{"className":36764},[304,305],[19,36766,36768,36807],{"className":36767},[137,138],[19,36769,36771,36804],{"className":36770},[142],[19,36772,36774,36794],{"className":36773,"style":315},[146],[19,36775,36776,36779],{"style":318},[19,36777],{"className":36778,"style":322},[154],[19,36780,36782],{"className":36781},[159,160,161,162],[19,36783,36785,36788,36791],{"className":36784},[75,162],[19,36786,3297],{"className":36787},[75,76,162],[19,36789,218],{"className":36790},[335,162],[19,36792,26855],{"className":36793},[75,162],[19,36795,36796,36799],{"style":387},[19,36797],{"className":36798,"style":322},[154],[19,36800,36801],{},[19,36802,208],{"className":36803},[304],[19,36805,173],{"className":36806},[172],[19,36808,36810],{"className":36809},[142],[19,36811,36813],{"className":36812,"style":35166},[146],[19,36814],{},[19,36816],{"className":36817,"style":412},[411],[19,36819,36821],{"className":36820},[75],[19,36822,36824,36827,36985],{"className":36823},[75],[19,36825],{"className":36826},[81,422],[19,36828,36830],{"className":36829},[226],[19,36831,36833,36977],{"className":36832},[137,138],[19,36834,36836,36974],{"className":36835},[142],[19,36837,36839,36899,36907],{"className":36838,"style":435},[146],[19,36840,36841,36844],{"style":438},[19,36842],{"className":36843,"style":322},[154],[19,36845,36847],{"className":36846},[75],[19,36848,36850,36853,36896],{"className":36849},[16026],[19,36851,29161],{"className":36852,"style":16031},[81,16030],[19,36854,36856,36859],{"className":36855},[75],[19,36857,3484],{"className":36858},[75,76],[19,36860,36862],{"className":36861},[133],[19,36863,36865,36888],{"className":36864},[137,138],[19,36866,36868,36885],{"className":36867},[142],[19,36869,36871],{"className":36870,"style":3609},[146],[19,36872,36873,36876],{"style":150},[19,36874],{"className":36875,"style":155},[154],[19,36877,36879],{"className":36878},[159,160,161,162],[19,36880,36882],{"className":36881},[75,162],[19,36883,3297],{"className":36884},[75,76,162],[19,36886,173],{"className":36887},[172],[19,36889,36891],{"className":36890},[142],[19,36892,36894],{"className":36893,"style":180},[146],[19,36895],{},[19,36897,29161],{"className":36898,"style":16031},[88,16030],[19,36900,36901,36904],{"style":504},[19,36902],{"className":36903,"style":322},[154],[19,36905],{"className":36906,"style":512},[511],[19,36908,36909,36912],{"style":515},[19,36910],{"className":36911,"style":322},[154],[19,36913,36915],{"className":36914},[75],[19,36916,36918,36921,36971],{"className":36917},[16026],[19,36919,29161],{"className":36920,"style":16031},[81,16030],[19,36922,36924,36927],{"className":36923},[75],[19,36925,3484],{"className":36926},[75,76],[19,36928,36930],{"className":36929},[133],[19,36931,36933,36962],{"className":36932},[137,138],[19,36934,36936,36959],{"className":36935},[142],[19,36937,36939],{"className":36938,"style":147},[146],[19,36940,36941,36944],{"style":150},[19,36942],{"className":36943,"style":155},[154],[19,36945,36947],{"className":36946},[159,160,161,162],[19,36948,36950,36953,36956],{"className":36949},[75,162],[19,36951,3297],{"className":36952},[75,76,162],[19,36954,2351],{"className":36955},[455,162],[19,36957,3495],{"className":36958},[75,162],[19,36960,173],{"className":36961},[172],[19,36963,36965],{"className":36964},[142],[19,36966,36969],{"className":36967,"style":36968},[146],"height:0.2083em;",[19,36970],{},[19,36972,29161],{"className":36973,"style":16031},[88,16030],[19,36975,173],{"className":36976},[172],[19,36978,36980],{"className":36979},[142],[19,36981,36983],{"className":36982,"style":11497},[146],[19,36984],{},[19,36986],{"className":36987},[88,422],[19,36989],{"className":36990,"style":613},[411],[19,36992,267],{"className":36993},[335],[19,36995],{"className":36996,"style":613},[411],[19,36998,37000,37003],{"className":36999},[66],[19,37001],{"className":37002,"style":34211},[70],[19,37004,33987],{"className":37005},[75,76],[15,37007,37008],{},"then necessarily",[19,37010,37012],{"className":37011,"translate":23},[187],[19,37013,37015,37062],{"className":37014,"translate":23},[22],[19,37016,37018],{"className":37017},[27],[29,37019,37020],{"xmlns":31,"display":196},[33,37021,37022,37060],{},[36,37023,37024,37040,37056,37058],{},[202,37025,37026,37032],{},[36,37027,37028,37030],{},[39,37029,208],{},[43,37031,211],{},[36,37033,37034,37036,37038],{},[39,37035,3297],{},[43,37037,218],{},[39,37039,26855],{"mathvariant":275},[33967,37041,37042,37054],{},[36,37043,37044,37046,37052],{},[43,37045,29161],{"fence":62},[103,37047,37048,37050],{},[39,37049,3484],{},[39,37051,3297],{},[43,37053,29161],{"fence":62},[39,37055,3297],{},[43,37057,267],{},[39,37059,33987],{},[54,37061,33990],{"encoding":56},[19,37063,37065,37261],{"className":37064,"ariaHidden":62},[61],[19,37066,37068,37071,37124,37127,37252,37255,37258],{"className":37067},[66],[19,37069],{"className":37070,"style":35114},[70],[19,37072,37074],{"className":37073},[304,305],[19,37075,37077,37116],{"className":37076},[137,138],[19,37078,37080,37113],{"className":37079},[142],[19,37081,37083,37103],{"className":37082,"style":315},[146],[19,37084,37085,37088],{"style":318},[19,37086],{"className":37087,"style":322},[154],[19,37089,37091],{"className":37090},[159,160,161,162],[19,37092,37094,37097,37100],{"className":37093},[75,162],[19,37095,3297],{"className":37096},[75,76,162],[19,37098,218],{"className":37099},[335,162],[19,37101,26855],{"className":37102},[75,162],[19,37104,37105,37108],{"style":387},[19,37106],{"className":37107,"style":322},[154],[19,37109,37110],{},[19,37111,208],{"className":37112},[304],[19,37114,173],{"className":37115},[172],[19,37117,37119],{"className":37118},[142],[19,37120,37122],{"className":37121,"style":35166},[146],[19,37123],{},[19,37125],{"className":37126,"style":412},[411],[19,37128,37130],{"className":37129},[75],[19,37131,37133,37159],{"className":37132},[75,34060],[19,37134,37136],{"className":37135},[34064],[19,37137,37139],{"className":37138},[137],[19,37140,37142],{"className":37141},[142],[19,37143,37145],{"className":37144,"style":35190},[146],[19,37146,37147,37150],{"style":35193},[19,37148],{"className":37149,"style":361},[154],[19,37151,37153],{"className":37152},[159,160,366,162],[19,37154,37156],{"className":37155},[75,162],[19,37157,3297],{"className":37158},[75,76,162],[19,37160,37162,37244],{"className":37161},[137,138],[19,37163,37165,37241],{"className":37164},[142],[19,37166,37168,37229],{"className":37167,"style":35215},[146],[19,37169,37171,37174],{"className":37170,"style":34104},[34103],[19,37172],{"className":37173,"style":30456},[154],[19,37175,37177],{"className":37176,"style":34111},[75],[19,37178,37180,37183,37226],{"className":37179},[16026],[19,37181,29161],{"className":37182,"style":16031},[81,16030],[19,37184,37186,37189],{"className":37185},[75],[19,37187,3484],{"className":37188},[75,76],[19,37190,37192],{"className":37191},[133],[19,37193,37195,37218],{"className":37194},[137,138],[19,37196,37198,37215],{"className":37197},[142],[19,37199,37201],{"className":37200,"style":3609},[146],[19,37202,37203,37206],{"style":150},[19,37204],{"className":37205,"style":155},[154],[19,37207,37209],{"className":37208},[159,160,161,162],[19,37210,37212],{"className":37211},[75,162],[19,37213,3297],{"className":37214},[75,76,162],[19,37216,173],{"className":37217},[172],[19,37219,37221],{"className":37220},[142],[19,37222,37224],{"className":37223,"style":180},[146],[19,37225],{},[19,37227,29161],{"className":37228,"style":16031},[88,16030],[19,37230,37231,37234],{"style":35311},[19,37232],{"className":37233,"style":30456},[154],[19,37235,37237],{"className":37236,"style":34174},[34173],[30461,37238,37239],{"xmlns":30463,"width":34177,"height":34178,"viewBox":34179,"preserveAspectRatio":34180},[30468,37240],{"d":34183},[19,37242,173],{"className":37243},[172],[19,37245,37247],{"className":37246},[142],[19,37248,37250],{"className":37249,"style":35331},[146],[19,37251],{},[19,37253],{"className":37254,"style":613},[411],[19,37256,267],{"className":37257},[335],[19,37259],{"className":37260,"style":613},[411],[19,37262,37264,37267],{"className":37263},[66],[19,37265],{"className":37266,"style":34211},[70],[19,37268,33987],{"className":37269},[75,76],[15,37271,37272,37273,37468,37469,37677],{},"Therefore, in finding the convergence radius, the ratio test can also be utilized. Additionally, in the above proof, we took the absolute value of each term of the original power series ",[19,37274,37276,37315],{"className":37275,"translate":23},[22],[19,37277,37279],{"className":37278},[27],[29,37280,37281],{"xmlns":31},[33,37282,37283,37313],{},[36,37284,37285,37299],{},[4285,37286,37287,37289,37297],{},[43,37288,26844],{},[36,37290,37291,37293,37295],{},[39,37292,3297],{},[43,37294,267],{},[108,37296,110],{},[39,37298,26855],{"mathvariant":275},[36,37300,37301,37307],{},[103,37302,37303,37305],{},[39,37304,3484],{},[39,37306,3297],{},[269,37308,37309,37311],{},[39,37310,49],{},[39,37312,3297],{},[54,37314,28591],{"encoding":56},[19,37316,37318],{"className":37317,"ariaHidden":62},[61],[19,37319,37321,37324,37387,37390],{"className":37320},[66],[19,37322],{"className":37323,"style":31929},[70],[19,37325,37327,37330],{"className":37326},[304],[19,37328,26844],{"className":37329,"style":31937},[304,27031,31936],[19,37331,37333],{"className":37332},[133],[19,37334,37336,37379],{"className":37335},[137,138],[19,37337,37339,37376],{"className":37338},[142],[19,37340,37342,37362],{"className":37341,"style":31950},[146],[19,37343,37344,37347],{"style":31953},[19,37345],{"className":37346,"style":155},[154],[19,37348,37350],{"className":37349},[159,160,161,162],[19,37351,37353,37356,37359],{"className":37352},[75,162],[19,37354,3297],{"className":37355},[75,76,162],[19,37357,267],{"className":37358},[335,162],[19,37360,110],{"className":37361},[75,162],[19,37363,37364,37367],{"style":31974},[19,37365],{"className":37366,"style":155},[154],[19,37368,37370],{"className":37369},[159,160,161,162],[19,37371,37373],{"className":37372},[75,162],[19,37374,26855],{"className":37375},[75,162],[19,37377,173],{"className":37378},[172],[19,37380,37382],{"className":37381},[142],[19,37383,37385],{"className":37384,"style":31996},[146],[19,37386],{},[19,37388],{"className":37389,"style":412},[411],[19,37391,37393,37436],{"className":37392},[75],[19,37394,37396,37399],{"className":37395},[75],[19,37397,3484],{"className":37398},[75,76],[19,37400,37402],{"className":37401},[133],[19,37403,37405,37428],{"className":37404},[137,138],[19,37406,37408,37425],{"className":37407},[142],[19,37409,37411],{"className":37410,"style":3609},[146],[19,37412,37413,37416],{"style":150},[19,37414],{"className":37415,"style":155},[154],[19,37417,37419],{"className":37418},[159,160,161,162],[19,37420,37422],{"className":37421},[75,162],[19,37423,3297],{"className":37424},[75,76,162],[19,37426,173],{"className":37427},[172],[19,37429,37431],{"className":37430},[142],[19,37432,37434],{"className":37433,"style":180},[146],[19,37435],{},[19,37437,37439,37442],{"className":37438},[75],[19,37440,49],{"className":37441},[75,76],[19,37443,37445],{"className":37444},[133],[19,37446,37448],{"className":37447},[137],[19,37449,37451],{"className":37450},[142],[19,37452,37454],{"className":37453,"style":3429},[146],[19,37455,37456,37459],{"style":1643},[19,37457],{"className":37458,"style":155},[154],[19,37460,37462],{"className":37461},[159,160,161,162],[19,37463,37465],{"className":37464},[75,162],[19,37466,3297],{"className":37467},[75,76,162]," to obtain the power series ",[19,37470,37472,37515],{"className":37471,"translate":23},[22],[19,37473,37475],{"className":37474},[27],[29,37476,37477],{"xmlns":31},[33,37478,37479,37513],{},[36,37480,37481,37495],{},[4285,37482,37483,37485,37493],{},[43,37484,26844],{},[36,37486,37487,37489,37491],{},[39,37488,3297],{},[43,37490,267],{},[108,37492,110],{},[39,37494,26855],{"mathvariant":275},[36,37496,37497,37499,37505,37511],{},[43,37498,29161],{"fence":62},[103,37500,37501,37503],{},[39,37502,3484],{},[39,37504,3297],{},[269,37506,37507,37509],{},[39,37508,49],{},[39,37510,3297],{},[43,37512,29161],{"fence":62},[54,37514,34824],{"encoding":56},[19,37516,37518],{"className":37517,"ariaHidden":62},[61],[19,37519,37521,37524,37587,37590],{"className":37520},[66],[19,37522],{"className":37523,"style":31929},[70],[19,37525,37527,37530],{"className":37526},[304],[19,37528,26844],{"className":37529,"style":31937},[304,27031,31936],[19,37531,37533],{"className":37532},[133],[19,37534,37536,37579],{"className":37535},[137,138],[19,37537,37539,37576],{"className":37538},[142],[19,37540,37542,37562],{"className":37541,"style":31950},[146],[19,37543,37544,37547],{"style":31953},[19,37545],{"className":37546,"style":155},[154],[19,37548,37550],{"className":37549},[159,160,161,162],[19,37551,37553,37556,37559],{"className":37552},[75,162],[19,37554,3297],{"className":37555},[75,76,162],[19,37557,267],{"className":37558},[335,162],[19,37560,110],{"className":37561},[75,162],[19,37563,37564,37567],{"style":31974},[19,37565],{"className":37566,"style":155},[154],[19,37568,37570],{"className":37569},[159,160,161,162],[19,37571,37573],{"className":37572},[75,162],[19,37574,26855],{"className":37575},[75,162],[19,37577,173],{"className":37578},[172],[19,37580,37582],{"className":37581},[142],[19,37583,37585],{"className":37584,"style":31996},[146],[19,37586],{},[19,37588],{"className":37589,"style":412},[411],[19,37591,37593],{"className":37592},[75],[19,37594,37596,37599,37642,37674],{"className":37595},[16026],[19,37597,29161],{"className":37598,"style":16031},[81,16030],[19,37600,37602,37605],{"className":37601},[75],[19,37603,3484],{"className":37604},[75,76],[19,37606,37608],{"className":37607},[133],[19,37609,37611,37634],{"className":37610},[137,138],[19,37612,37614,37631],{"className":37613},[142],[19,37615,37617],{"className":37616,"style":3609},[146],[19,37618,37619,37622],{"style":150},[19,37620],{"className":37621,"style":155},[154],[19,37623,37625],{"className":37624},[159,160,161,162],[19,37626,37628],{"className":37627},[75,162],[19,37629,3297],{"className":37630},[75,76,162],[19,37632,173],{"className":37633},[172],[19,37635,37637],{"className":37636},[142],[19,37638,37640],{"className":37639,"style":180},[146],[19,37641],{},[19,37643,37645,37648],{"className":37644},[75],[19,37646,49],{"className":37647},[75,76],[19,37649,37651],{"className":37650},[133],[19,37652,37654],{"className":37653},[137],[19,37655,37657],{"className":37656},[142],[19,37658,37660],{"className":37659,"style":3429},[146],[19,37661,37662,37665],{"style":1643},[19,37663],{"className":37664,"style":155},[154],[19,37666,37668],{"className":37667},[159,160,161,162],[19,37669,37671],{"className":37670},[75,162],[19,37672,3297],{"className":37673},[75,76,162],[19,37675,29161],{"className":37676,"style":16031},[88,16030]," for demonstration. Since the power series is absolutely convergent within its convergence interval, these two power series have exactly the same convergence radius. Thus, this proof method is effective.",[15,37679,37680,37681,37914,37915,38110],{},"Certainly, the method mentioned above for finding the convergence radius has limitations. If the limit ",[19,37682,37684,37728],{"className":37683,"translate":23},[22],[19,37685,37687],{"className":37686},[27],[29,37688,37689],{"xmlns":31},[33,37690,37691,37725],{},[36,37692,37693,37709],{},[103,37694,37695,37701],{},[36,37696,37697,37699],{},[39,37698,208],{},[43,37700,211],{},[36,37702,37703,37705,37707],{},[39,37704,3297],{},[43,37706,218],{},[39,37708,26855],{"mathvariant":275},[33967,37710,37711,37723],{},[36,37712,37713,37715,37721],{},[43,37714,29161],{"fence":62},[103,37716,37717,37719],{},[39,37718,3484],{},[39,37720,3297],{},[43,37722,29161],{"fence":62},[39,37724,3297],{},[54,37726,37727],{"encoding":56},"\\lim_{n \\to \\infty}{\\sqrt[n]{\\left| a_{n} \\right|}}",[19,37729,37731],{"className":37730,"ariaHidden":62},[61],[19,37732,37734,37737,37786,37789],{"className":37733},[66],[19,37735],{"className":37736,"style":34000},[70],[19,37738,37740,37743],{"className":37739},[304],[19,37741,208],{"className":37742},[304],[19,37744,37746],{"className":37745},[133],[19,37747,37749,37778],{"className":37748},[137,138],[19,37750,37752,37775],{"className":37751},[142],[19,37753,37755],{"className":37754,"style":3609},[146],[19,37756,37757,37760],{"style":34021},[19,37758],{"className":37759,"style":155},[154],[19,37761,37763],{"className":37762},[159,160,161,162],[19,37764,37766,37769,37772],{"className":37765},[75,162],[19,37767,3297],{"className":37768},[75,76,162],[19,37770,218],{"className":37771},[335,162],[19,37773,26855],{"className":37774},[75,162],[19,37776,173],{"className":37777},[172],[19,37779,37781],{"className":37780},[142],[19,37782,37784],{"className":37783,"style":180},[146],[19,37785],{},[19,37787],{"className":37788,"style":412},[411],[19,37790,37792],{"className":37791},[75],[19,37793,37795,37821],{"className":37794},[75,34060],[19,37796,37798],{"className":37797},[34064],[19,37799,37801],{"className":37800},[137],[19,37802,37804],{"className":37803},[142],[19,37805,37807],{"className":37806,"style":34074},[146],[19,37808,37809,37812],{"style":34077},[19,37810],{"className":37811,"style":361},[154],[19,37813,37815],{"className":37814},[159,160,366,162],[19,37816,37818],{"className":37817},[75,162],[19,37819,3297],{"className":37820},[75,76,162],[19,37822,37824,37906],{"className":37823},[137,138],[19,37825,37827,37903],{"className":37826},[142],[19,37828,37830,37891],{"className":37829,"style":34099},[146],[19,37831,37833,37836],{"className":37832,"style":34104},[34103],[19,37834],{"className":37835,"style":30456},[154],[19,37837,37839],{"className":37838,"style":34111},[75],[19,37840,37842,37845,37888],{"className":37841},[16026],[19,37843,29161],{"className":37844,"style":16031},[81,16030],[19,37846,37848,37851],{"className":37847},[75],[19,37849,3484],{"className":37850},[75,76],[19,37852,37854],{"className":37853},[133],[19,37855,37857,37880],{"className":37856},[137,138],[19,37858,37860,37877],{"className":37859},[142],[19,37861,37863],{"className":37862,"style":3609},[146],[19,37864,37865,37868],{"style":150},[19,37866],{"className":37867,"style":155},[154],[19,37869,37871],{"className":37870},[159,160,161,162],[19,37872,37874],{"className":37873},[75,162],[19,37875,3297],{"className":37876},[75,76,162],[19,37878,173],{"className":37879},[172],[19,37881,37883],{"className":37882},[142],[19,37884,37886],{"className":37885,"style":180},[146],[19,37887],{},[19,37889,29161],{"className":37890,"style":16031},[88,16030],[19,37892,37893,37896],{"style":34166},[19,37894],{"className":37895,"style":30456},[154],[19,37897,37899],{"className":37898,"style":34174},[34173],[30461,37900,37901],{"xmlns":30463,"width":34177,"height":34178,"viewBox":34179,"preserveAspectRatio":34180},[30468,37902],{"d":34183},[19,37904,173],{"className":37905},[172],[19,37907,37909],{"className":37908},[142],[19,37910,37912],{"className":37911,"style":34193},[146],[19,37913],{}," does not exist (and is not positive infinity), then this method will fail. Therefore, in general mathematical analysis textbooks, a more general method for finding the convergence radius of a power series ",[19,37916,37918,37957],{"className":37917,"translate":23},[22],[19,37919,37921],{"className":37920},[27],[29,37922,37923],{"xmlns":31},[33,37924,37925,37955],{},[36,37926,37927,37941],{},[4285,37928,37929,37931,37939],{},[43,37930,26844],{},[36,37932,37933,37935,37937],{},[39,37934,3297],{},[43,37936,267],{},[108,37938,110],{},[39,37940,26855],{"mathvariant":275},[36,37942,37943,37949],{},[103,37944,37945,37947],{},[39,37946,3484],{},[39,37948,3297],{},[269,37950,37951,37953],{},[39,37952,49],{},[39,37954,3297],{},[54,37956,28591],{"encoding":56},[19,37958,37960],{"className":37959,"ariaHidden":62},[61],[19,37961,37963,37966,38029,38032],{"className":37962},[66],[19,37964],{"className":37965,"style":31929},[70],[19,37967,37969,37972],{"className":37968},[304],[19,37970,26844],{"className":37971,"style":31937},[304,27031,31936],[19,37973,37975],{"className":37974},[133],[19,37976,37978,38021],{"className":37977},[137,138],[19,37979,37981,38018],{"className":37980},[142],[19,37982,37984,38004],{"className":37983,"style":31950},[146],[19,37985,37986,37989],{"style":31953},[19,37987],{"className":37988,"style":155},[154],[19,37990,37992],{"className":37991},[159,160,161,162],[19,37993,37995,37998,38001],{"className":37994},[75,162],[19,37996,3297],{"className":37997},[75,76,162],[19,37999,267],{"className":38000},[335,162],[19,38002,110],{"className":38003},[75,162],[19,38005,38006,38009],{"style":31974},[19,38007],{"className":38008,"style":155},[154],[19,38010,38012],{"className":38011},[159,160,161,162],[19,38013,38015],{"className":38014},[75,162],[19,38016,26855],{"className":38017},[75,162],[19,38019,173],{"className":38020},[172],[19,38022,38024],{"className":38023},[142],[19,38025,38027],{"className":38026,"style":31996},[146],[19,38028],{},[19,38030],{"className":38031,"style":412},[411],[19,38033,38035,38078],{"className":38034},[75],[19,38036,38038,38041],{"className":38037},[75],[19,38039,3484],{"className":38040},[75,76],[19,38042,38044],{"className":38043},[133],[19,38045,38047,38070],{"className":38046},[137,138],[19,38048,38050,38067],{"className":38049},[142],[19,38051,38053],{"className":38052,"style":3609},[146],[19,38054,38055,38058],{"style":150},[19,38056],{"className":38057,"style":155},[154],[19,38059,38061],{"className":38060},[159,160,161,162],[19,38062,38064],{"className":38063},[75,162],[19,38065,3297],{"className":38066},[75,76,162],[19,38068,173],{"className":38069},[172],[19,38071,38073],{"className":38072},[142],[19,38074,38076],{"className":38075,"style":180},[146],[19,38077],{},[19,38079,38081,38084],{"className":38080},[75],[19,38082,49],{"className":38083},[75,76],[19,38085,38087],{"className":38086},[133],[19,38088,38090],{"className":38089},[137],[19,38091,38093],{"className":38092},[142],[19,38094,38096],{"className":38095,"style":3429},[146],[19,38097,38098,38101],{"style":1643},[19,38099],{"className":38100,"style":155},[154],[19,38102,38104],{"className":38103},[159,160,161,162],[19,38105,38107],{"className":38106},[75,162],[19,38108,3297],{"className":38109},[75,76,162]," is given using the method of upper limits. Below is the theorem:",[15,38112,38113,33732],{},[7600,38114,38115],{},"(Extended) Theorem:",[19,38117,38119],{"className":38118,"translate":23},[187],[19,38120,38122,38161],{"className":38121,"translate":23},[22],[19,38123,38125],{"className":38124},[27],[29,38126,38127],{"xmlns":31,"display":196},[33,38128,38129,38159],{},[36,38130,38131,38145],{},[26840,38132,38133,38135,38143],{},[43,38134,26844],{},[36,38136,38137,38139,38141],{},[39,38138,3297],{},[43,38140,267],{},[108,38142,110],{},[39,38144,26855],{"mathvariant":275},[36,38146,38147,38153],{},[103,38148,38149,38151],{},[39,38150,3484],{},[39,38152,3297],{},[269,38154,38155,38157],{},[39,38156,49],{},[39,38158,3297],{},[54,38160,28591],{"encoding":56},[19,38162,38164],{"className":38163,"ariaHidden":62},[61],[19,38165,38167,38170,38237,38240],{"className":38166},[66],[19,38168],{"className":38169,"style":26984},[70],[19,38171,38173],{"className":38172},[304,305],[19,38174,38176,38229],{"className":38175},[137,138],[19,38177,38179,38226],{"className":38178},[142],[19,38180,38182,38202,38212],{"className":38181,"style":26997},[146],[19,38183,38184,38187],{"style":27000},[19,38185],{"className":38186,"style":27004},[154],[19,38188,38190],{"className":38189},[159,160,161,162],[19,38191,38193,38196,38199],{"className":38192},[75,162],[19,38194,3297],{"className":38195},[75,76,162],[19,38197,267],{"className":38198},[335,162],[19,38200,110],{"className":38201},[75,162],[19,38203,38204,38207],{"style":27022},[19,38205],{"className":38206,"style":27004},[154],[19,38208,38209],{},[19,38210,26844],{"className":38211},[304,27031,27032],[19,38213,38214,38217],{"style":27035},[19,38215],{"className":38216,"style":27004},[154],[19,38218,38220],{"className":38219},[159,160,161,162],[19,38221,38223],{"className":38222},[75,162],[19,38224,26855],{"className":38225},[75,162],[19,38227,173],{"className":38228},[172],[19,38230,38232],{"className":38231},[142],[19,38233,38235],{"className":38234,"style":27057},[146],[19,38236],{},[19,38238],{"className":38239,"style":412},[411],[19,38241,38243,38286],{"className":38242},[75],[19,38244,38246,38249],{"className":38245},[75],[19,38247,3484],{"className":38248},[75,76],[19,38250,38252],{"className":38251},[133],[19,38253,38255,38278],{"className":38254},[137,138],[19,38256,38258,38275],{"className":38257},[142],[19,38259,38261],{"className":38260,"style":3609},[146],[19,38262,38263,38266],{"style":150},[19,38264],{"className":38265,"style":155},[154],[19,38267,38269],{"className":38268},[159,160,161,162],[19,38270,38272],{"className":38271},[75,162],[19,38273,3297],{"className":38274},[75,76,162],[19,38276,173],{"className":38277},[172],[19,38279,38281],{"className":38280},[142],[19,38282,38284],{"className":38283,"style":180},[146],[19,38285],{},[19,38287,38289,38292],{"className":38288},[75],[19,38290,49],{"className":38291},[75,76],[19,38293,38295],{"className":38294},[133],[19,38296,38298],{"className":38297},[137],[19,38299,38301],{"className":38300},[142],[19,38302,38304],{"className":38303,"style":4148},[146],[19,38305,38306,38309],{"style":648},[19,38307],{"className":38308,"style":155},[154],[19,38310,38312],{"className":38311},[159,160,161,162],[19,38313,38315],{"className":38314},[75,162],[19,38316,3297],{"className":38317},[75,76,162],[15,38319,38320],{},"If",[19,38322,38324],{"className":38323,"translate":23},[187],[19,38325,38327,38390],{"className":38326,"translate":23},[22],[19,38328,38330],{"className":38329},[27],[29,38331,38332],{"xmlns":31,"display":196},[33,38333,38334,38387],{},[36,38335,38336,38367,38383,38385],{},[202,38337,38338,38359],{},[36,38339,38340,38357],{},[39,38341,38342],{"mathvariant":275},[28980,38343,38344,38354],{"accent":62},[36,38345,38346,38349,38351],{},[39,38347,38348],{"mathvariant":275},"l",[39,38350,34238],{"mathvariant":275},[39,38352,38353],{"mathvariant":275},"m",[43,38355,38356],{"stretchy":62},"‾",[43,38358,211],{},[36,38360,38361,38363,38365],{},[39,38362,3297],{},[43,38364,218],{},[39,38366,26855],{"mathvariant":275},[33967,38368,38369,38381],{},[36,38370,38371,38373,38379],{},[43,38372,29161],{"fence":62},[103,38374,38375,38377],{},[39,38376,3484],{},[39,38378,3297],{},[43,38380,29161],{"fence":62},[39,38382,3297],{},[43,38384,267],{},[39,38386,33987],{},[54,38388,38389],{"encoding":56},"\\varlimsup _{n \\rightarrow \\infty}{\\sqrt[n]{\\left| a_{n} \\right|}}=\\rho",[19,38391,38393,38625],{"className":38392,"ariaHidden":62},[61],[19,38394,38396,38399,38488,38491,38616,38619,38622],{"className":38395},[66],[19,38397],{"className":38398,"style":35114},[70],[19,38400,38402],{"className":38401},[304,305],[19,38403,38405,38480],{"className":38404},[137,138],[19,38406,38408,38477],{"className":38407},[142],[19,38409,38412,38432],{"className":38410,"style":38411},[146],"height:0.8944em;",[19,38413,38414,38417],{"style":318},[19,38415],{"className":38416,"style":322},[154],[19,38418,38420],{"className":38419},[159,160,161,162],[19,38421,38423,38426,38429],{"className":38422},[75,162],[19,38424,3297],{"className":38425},[75,76,162],[19,38427,218],{"className":38428},[335,162],[19,38430,26855],{"className":38431},[75,162],[19,38433,38434,38437],{"style":387},[19,38435],{"className":38436,"style":322},[154],[19,38438,38439],{},[19,38440,38442],{"className":38441},[304],[19,38443,38446],{"className":38444},[75,38445],"overline",[19,38447,38449],{"className":38448},[137],[19,38450,38452],{"className":38451},[142],[19,38453,38455,38467],{"className":38454,"style":38411},[146],[19,38456,38457,38460],{"style":387},[19,38458],{"className":38459,"style":322},[154],[19,38461,38463],{"className":38462},[75],[19,38464,208],{"className":38465},[75,38466],"mathrm",[19,38468,38470,38473],{"style":38469},"top:-3.8144em;",[19,38471],{"className":38472,"style":322},[154],[19,38474],{"className":38475,"style":512},[38476],"overline-line",[19,38478,173],{"className":38479},[172],[19,38481,38483],{"className":38482},[142],[19,38484,38486],{"className":38485,"style":35166},[146],[19,38487],{},[19,38489],{"className":38490,"style":412},[411],[19,38492,38494],{"className":38493},[75],[19,38495,38497,38523],{"className":38496},[75,34060],[19,38498,38500],{"className":38499},[34064],[19,38501,38503],{"className":38502},[137],[19,38504,38506],{"className":38505},[142],[19,38507,38509],{"className":38508,"style":35190},[146],[19,38510,38511,38514],{"style":35193},[19,38512],{"className":38513,"style":361},[154],[19,38515,38517],{"className":38516},[159,160,366,162],[19,38518,38520],{"className":38519},[75,162],[19,38521,3297],{"className":38522},[75,76,162],[19,38524,38526,38608],{"className":38525},[137,138],[19,38527,38529,38605],{"className":38528},[142],[19,38530,38532,38593],{"className":38531,"style":35215},[146],[19,38533,38535,38538],{"className":38534,"style":34104},[34103],[19,38536],{"className":38537,"style":30456},[154],[19,38539,38541],{"className":38540,"style":34111},[75],[19,38542,38544,38547,38590],{"className":38543},[16026],[19,38545,29161],{"className":38546,"style":16031},[81,16030],[19,38548,38550,38553],{"className":38549},[75],[19,38551,3484],{"className":38552},[75,76],[19,38554,38556],{"className":38555},[133],[19,38557,38559,38582],{"className":38558},[137,138],[19,38560,38562,38579],{"className":38561},[142],[19,38563,38565],{"className":38564,"style":3609},[146],[19,38566,38567,38570],{"style":150},[19,38568],{"className":38569,"style":155},[154],[19,38571,38573],{"className":38572},[159,160,161,162],[19,38574,38576],{"className":38575},[75,162],[19,38577,3297],{"className":38578},[75,76,162],[19,38580,173],{"className":38581},[172],[19,38583,38585],{"className":38584},[142],[19,38586,38588],{"className":38587,"style":180},[146],[19,38589],{},[19,38591,29161],{"className":38592,"style":16031},[88,16030],[19,38594,38595,38598],{"style":35311},[19,38596],{"className":38597,"style":30456},[154],[19,38599,38601],{"className":38600,"style":34174},[34173],[30461,38602,38603],{"xmlns":30463,"width":34177,"height":34178,"viewBox":34179,"preserveAspectRatio":34180},[30468,38604],{"d":34183},[19,38606,173],{"className":38607},[172],[19,38609,38611],{"className":38610},[142],[19,38612,38614],{"className":38613,"style":35331},[146],[19,38615],{},[19,38617],{"className":38618,"style":613},[411],[19,38620,267],{"className":38621},[335],[19,38623],{"className":38624,"style":613},[411],[19,38626,38628,38631],{"className":38627},[66],[19,38629],{"className":38630,"style":34211},[70],[19,38632,33987],{"className":38633},[75,76],[15,38635,38636],{},"then when",[34217,38638,38639,38875,39022],{},[34220,38640,38641,34262,38679,34342,38756,34466],{},[19,38642,38644,38661],{"className":38643,"translate":23},[22],[19,38645,38647],{"className":38646},[27],[29,38648,38649],{"xmlns":31},[33,38650,38651,38659],{},[36,38652,38653,38655,38657],{},[43,38654,46],{"stretchy":45},[39,38656,34238],{},[43,38658,52],{"stretchy":45},[54,38660,34243],{"encoding":56},[19,38662,38664],{"className":38663,"ariaHidden":62},[61],[19,38665,38667,38670,38673,38676],{"className":38666},[66],[19,38668],{"className":38669,"style":71},[70],[19,38671,46],{"className":38672},[81],[19,38674,34238],{"className":38675},[75,76],[19,38677,52],{"className":38678},[88],[19,38680,38682,38705],{"className":38681,"translate":23},[22],[19,38683,38685],{"className":38684},[27],[29,38686,38687],{"xmlns":31},[33,38688,38689,38703],{},[36,38690,38691,38693,38695,38697,38699,38701],{},[108,38692,110],{},[43,38694,29168],{},[39,38696,33987],{},[43,38698,29168],{},[43,38700,2351],{},[39,38702,26855],{"mathvariant":275},[54,38704,34289],{"encoding":56},[19,38706,38708,38726,38744],{"className":38707,"ariaHidden":62},[61],[19,38709,38711,38714,38717,38720,38723],{"className":38710},[66],[19,38712],{"className":38713,"style":34299},[70],[19,38715,110],{"className":38716},[75],[19,38718],{"className":38719,"style":613},[411],[19,38721,29168],{"className":38722},[335],[19,38724],{"className":38725,"style":613},[411],[19,38727,38729,38732,38735,38738,38741],{"className":38728},[66],[19,38730],{"className":38731,"style":17418},[70],[19,38733,33987],{"className":38734},[75,76],[19,38736],{"className":38737,"style":613},[411],[19,38739,29168],{"className":38740},[335],[19,38742],{"className":38743,"style":613},[411],[19,38745,38747,38750,38753],{"className":38746},[66],[19,38748],{"className":38749,"style":4003},[70],[19,38751,2351],{"className":38752},[75],[19,38754,26855],{"className":38755},[75],[19,38757,38759,38780],{"className":38758,"translate":23},[22],[19,38760,38762],{"className":38761},[27],[29,38763,38764],{"xmlns":31},[33,38765,38766,38778],{},[36,38767,38768,38770,38772],{},[39,38769,10758],{},[43,38771,267],{},[226,38773,38774,38776],{},[108,38775,3495],{},[39,38777,33987],{},[54,38779,34367],{"encoding":56},[19,38781,38783,38801],{"className":38782,"ariaHidden":62},[61],[19,38784,38786,38789,38792,38795,38798],{"className":38785},[66],[19,38787],{"className":38788,"style":7738},[70],[19,38790,10758],{"className":38791,"style":10844},[75,76],[19,38793],{"className":38794,"style":613},[411],[19,38796,267],{"className":38797},[335],[19,38799],{"className":38800,"style":613},[411],[19,38802,38804,38807],{"className":38803},[66],[19,38805],{"className":38806,"style":34395},[70],[19,38808,38810,38813,38872],{"className":38809},[75],[19,38811],{"className":38812},[81,422],[19,38814,38816],{"className":38815},[226],[19,38817,38819,38864],{"className":38818},[137,138],[19,38820,38822,38861],{"className":38821},[142],[19,38823,38825,38839,38847],{"className":38824,"style":34414},[146],[19,38826,38827,38830],{"style":30504},[19,38828],{"className":38829,"style":322},[154],[19,38831,38833],{"className":38832},[159,160,161,162],[19,38834,38836],{"className":38835},[75,162],[19,38837,33987],{"className":38838},[75,76,162],[19,38840,38841,38844],{"style":504},[19,38842],{"className":38843,"style":322},[154],[19,38845],{"className":38846,"style":512},[511],[19,38848,38849,38852],{"style":30556},[19,38850],{"className":38851,"style":322},[154],[19,38853,38855],{"className":38854},[159,160,161,162],[19,38856,38858],{"className":38857},[75,162],[19,38859,3495],{"className":38860},[75,162],[19,38862,173],{"className":38863},[172],[19,38865,38867],{"className":38866},[142],[19,38868,38870],{"className":38869,"style":34460},[146],[19,38871],{},[19,38873],{"className":38874},[88,422],[34220,38876,38877,34262,38917,34342,38967,34466],{},[19,38878,38880,38899],{"className":38879,"translate":23},[22],[19,38881,38883],{"className":38882},[27],[29,38884,38885],{"xmlns":31},[33,38886,38887,38897],{},[36,38888,38889,38891,38893,38895],{},[43,38890,46],{"stretchy":45},[39,38892,34238],{},[39,38894,34238],{},[43,38896,52],{"stretchy":45},[54,38898,34491],{"encoding":56},[19,38900,38902],{"className":38901,"ariaHidden":62},[61],[19,38903,38905,38908,38911,38914],{"className":38904},[66],[19,38906],{"className":38907,"style":71},[70],[19,38909,46],{"className":38910},[81],[19,38912,34507],{"className":38913},[75,76],[19,38915,52],{"className":38916},[88],[19,38918,38920,38937],{"className":38919,"translate":23},[22],[19,38921,38923],{"className":38922},[27],[29,38924,38925],{"xmlns":31},[33,38926,38927,38935],{},[36,38928,38929,38931,38933],{},[39,38930,33987],{},[43,38932,267],{},[108,38934,110],{},[54,38936,34531],{"encoding":56},[19,38938,38940,38958],{"className":38939,"ariaHidden":62},[61],[19,38941,38943,38946,38949,38952,38955],{"className":38942},[66],[19,38944],{"className":38945,"style":34211},[70],[19,38947,33987],{"className":38948},[75,76],[19,38950],{"className":38951,"style":613},[411],[19,38953,267],{"className":38954},[335],[19,38956],{"className":38957,"style":613},[411],[19,38959,38961,38964],{"className":38960},[66],[19,38962],{"className":38963,"style":1214},[70],[19,38965,110],{"className":38966},[75],[19,38968,38970,38989],{"className":38969,"translate":23},[22],[19,38971,38973],{"className":38972},[27],[29,38974,38975],{"xmlns":31},[33,38976,38977,38987],{},[36,38978,38979,38981,38983,38985],{},[39,38980,10758],{},[43,38982,267],{},[43,38984,2351],{},[39,38986,26855],{"mathvariant":275},[54,38988,34584],{"encoding":56},[19,38990,38992,39010],{"className":38991,"ariaHidden":62},[61],[19,38993,38995,38998,39001,39004,39007],{"className":38994},[66],[19,38996],{"className":38997,"style":7738},[70],[19,38999,10758],{"className":39000,"style":10844},[75,76],[19,39002],{"className":39003,"style":613},[411],[19,39005,267],{"className":39006},[335],[19,39008],{"className":39009,"style":613},[411],[19,39011,39013,39016,39019],{"className":39012},[66],[19,39014],{"className":39015,"style":4003},[70],[19,39017,2351],{"className":39018},[75],[19,39020,26855],{"className":39021},[75],[34220,39023,39024,34262,39066,34342,39121,3447],{},[19,39025,39027,39048],{"className":39026,"translate":23},[22],[19,39028,39030],{"className":39029},[27],[29,39031,39032],{"xmlns":31},[33,39033,39034,39046],{},[36,39035,39036,39038,39040,39042,39044],{},[43,39037,46],{"stretchy":45},[39,39039,34238],{},[39,39041,34238],{},[39,39043,34238],{},[43,39045,52],{"stretchy":45},[54,39047,34644],{"encoding":56},[19,39049,39051],{"className":39050,"ariaHidden":62},[61],[19,39052,39054,39057,39060,39063],{"className":39053},[66],[19,39055],{"className":39056,"style":71},[70],[19,39058,46],{"className":39059},[81],[19,39061,34660],{"className":39062},[75,76],[19,39064,52],{"className":39065},[88],[19,39067,39069,39088],{"className":39068,"translate":23},[22],[19,39070,39072],{"className":39071},[27],[29,39073,39074],{"xmlns":31},[33,39075,39076,39086],{},[36,39077,39078,39080,39082,39084],{},[39,39079,15],{},[43,39081,267],{},[43,39083,2351],{},[39,39085,26855],{"mathvariant":275},[54,39087,34686],{"encoding":56},[19,39089,39091,39109],{"className":39090,"ariaHidden":62},[61],[19,39092,39094,39097,39100,39103,39106],{"className":39093},[66],[19,39095],{"className":39096,"style":34211},[70],[19,39098,15],{"className":39099},[75,76],[19,39101],{"className":39102,"style":613},[411],[19,39104,267],{"className":39105},[335],[19,39107],{"className":39108,"style":613},[411],[19,39110,39112,39115,39118],{"className":39111},[66],[19,39113],{"className":39114,"style":4003},[70],[19,39116,2351],{"className":39117},[75],[19,39119,26855],{"className":39120},[75],[19,39122,39124,39141],{"className":39123,"translate":23},[22],[19,39125,39127],{"className":39126},[27],[29,39128,39129],{"xmlns":31},[33,39130,39131,39139],{},[36,39132,39133,39135,39137],{},[39,39134,10758],{},[43,39136,267],{},[108,39138,110],{},[54,39140,34740],{"encoding":56},[19,39142,39144,39162],{"className":39143,"ariaHidden":62},[61],[19,39145,39147,39150,39153,39156,39159],{"className":39146},[66],[19,39148],{"className":39149,"style":7738},[70],[19,39151,10758],{"className":39152,"style":10844},[75,76],[19,39154],{"className":39155,"style":613},[411],[19,39157,267],{"className":39158},[335],[19,39160],{"className":39161,"style":613},[411],[19,39163,39165,39168],{"className":39164},[66],[19,39166],{"className":39167,"style":1214},[70],[19,39169,110],{"className":39170},[75],[15,39172,39173],{},"Since this upper limit always exists, any power series can be used to find the convergence radius using this theorem.",[15,39175,39176],{},"In practical problems of finding the convergence radius of power series, we may encounter \"missing term\" power series, such as the power series",[19,39178,39180],{"className":39179,"translate":23},[187],[19,39181,39183,39238],{"className":39182,"translate":23},[22],[19,39184,39186],{"className":39185},[27],[29,39187,39188],{"xmlns":31,"display":196},[33,39189,39190,39235],{},[36,39191,39192,39206],{},[26840,39193,39194,39196,39204],{},[43,39195,26844],{},[36,39197,39198,39200,39202],{},[39,39199,3297],{},[43,39201,267],{},[108,39203,3495],{},[39,39205,26855],{"mathvariant":275},[226,39207,39208,39218],{},[269,39209,39210,39212],{},[39,39211,49],{},[36,39213,39214,39216],{},[108,39215,3518],{},[39,39217,3297],{},[36,39219,39220,39222,39224],{},[39,39221,3297],{},[43,39223,240],{},[269,39225,39226,39229],{},[108,39227,39228],{},"3",[36,39230,39231,39233],{},[108,39232,3518],{},[39,39234,3297],{},[54,39236,39237],{"encoding":56},"\\sum_{n=1}^{\\infty}{\\frac{x^{2n}}{n-3^{2n}}}",[19,39239,39241],{"className":39240,"ariaHidden":62},[61],[19,39242,39244,39247,39314,39317],{"className":39243},[66],[19,39245],{"className":39246,"style":26984},[70],[19,39248,39250],{"className":39249},[304,305],[19,39251,39253,39306],{"className":39252},[137,138],[19,39254,39256,39303],{"className":39255},[142],[19,39257,39259,39279,39289],{"className":39258,"style":26997},[146],[19,39260,39261,39264],{"style":27000},[19,39262],{"className":39263,"style":27004},[154],[19,39265,39267],{"className":39266},[159,160,161,162],[19,39268,39270,39273,39276],{"className":39269},[75,162],[19,39271,3297],{"className":39272},[75,76,162],[19,39274,267],{"className":39275},[335,162],[19,39277,3495],{"className":39278},[75,162],[19,39280,39281,39284],{"style":27022},[19,39282],{"className":39283,"style":27004},[154],[19,39285,39286],{},[19,39287,26844],{"className":39288},[304,27031,27032],[19,39290,39291,39294],{"style":27035},[19,39292],{"className":39293,"style":27004},[154],[19,39295,39297],{"className":39296},[159,160,161,162],[19,39298,39300],{"className":39299},[75,162],[19,39301,26855],{"className":39302},[75,162],[19,39304,173],{"className":39305},[172],[19,39307,39309],{"className":39308},[142],[19,39310,39312],{"className":39311,"style":27057},[146],[19,39313],{},[19,39315],{"className":39316,"style":412},[411],[19,39318,39320],{"className":39319},[75],[19,39321,39323,39326,39458],{"className":39322},[75],[19,39324],{"className":39325},[81,422],[19,39327,39329],{"className":39328},[226],[19,39330,39332,39449],{"className":39331},[137,138],[19,39333,39335,39446],{"className":39334},[142],[19,39336,39339,39395,39403],{"className":39337,"style":39338},[146],"height:1.4911em;",[19,39340,39341,39344],{"style":438},[19,39342],{"className":39343,"style":322},[154],[19,39345,39347,39350,39353,39356,39359],{"className":39346},[75],[19,39348,3297],{"className":39349},[75,76],[19,39351],{"className":39352,"style":451},[411],[19,39354,240],{"className":39355},[455],[19,39357],{"className":39358,"style":451},[411],[19,39360,39362,39365],{"className":39361},[75],[19,39363,39228],{"className":39364},[75],[19,39366,39368],{"className":39367},[133],[19,39369,39371],{"className":39370},[137],[19,39372,39374],{"className":39373},[142],[19,39375,39378],{"className":39376,"style":39377},[146],"height:0.7401em;",[19,39379,39380,39383],{"style":14745},[19,39381],{"className":39382,"style":155},[154],[19,39384,39386],{"className":39385},[159,160,161,162],[19,39387,39389,39392],{"className":39388},[75,162],[19,39390,3518],{"className":39391},[75,162],[19,39393,3297],{"className":39394},[75,76,162],[19,39396,39397,39400],{"style":504},[19,39398],{"className":39399,"style":322},[154],[19,39401],{"className":39402,"style":512},[511],[19,39404,39405,39408],{"style":515},[19,39406],{"className":39407,"style":322},[154],[19,39409,39411],{"className":39410},[75],[19,39412,39414,39417],{"className":39413},[75],[19,39415,49],{"className":39416},[75,76],[19,39418,39420],{"className":39419},[133],[19,39421,39423],{"className":39422},[137],[19,39424,39426],{"className":39425},[142],[19,39427,39429],{"className":39428,"style":19038},[146],[19,39430,39431,39434],{"style":1643},[19,39432],{"className":39433,"style":155},[154],[19,39435,39437],{"className":39436},[159,160,161,162],[19,39438,39440,39443],{"className":39439},[75,162],[19,39441,3518],{"className":39442},[75,162],[19,39444,3297],{"className":39445},[75,76,162],[19,39447,173],{"className":39448},[172],[19,39450,39452],{"className":39451},[142],[19,39453,39456],{"className":39454,"style":39455},[146],"height:0.7693em;",[19,39457],{},[19,39459],{"className":39460},[88,422],[15,39462,39463],{},"Such power series can certainly be found using the method of upper limits mentioned above, but a more effective method is to use the root test to deduce the convergence radius of the power 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3",[19,40624,40626,40653],{"className":40625,"ariaHidden":62},[61],[19,40627,40629,40632,40644,40647,40650],{"className":40628},[66],[19,40630],{"className":40631,"style":71},[70],[19,40633,40635,40638,40641],{"className":40634},[16026],[19,40636,29161],{"className":40637,"style":16031},[81,16030],[19,40639,49],{"className":40640},[75,76],[19,40642,29161],{"className":40643,"style":16031},[88,16030],[19,40645],{"className":40646,"style":613},[411],[19,40648,29168],{"className":40649},[335],[19,40651],{"className":40652,"style":613},[411],[19,40654,40656,40659],{"className":40655},[66],[19,40657],{"className":40658,"style":1214},[70],[19,40660,39228],{"className":40661},[75],", the power series converges, and when 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x \\right| > 3",[19,40844,40846,40873],{"className":40845,"ariaHidden":62},[61],[19,40847,40849,40852,40864,40867,40870],{"className":40848},[66],[19,40850],{"className":40851,"style":71},[70],[19,40853,40855,40858,40861],{"className":40854},[16026],[19,40856,29161],{"className":40857,"style":16031},[81,16030],[19,40859,49],{"className":40860},[75,76],[19,40862,29161],{"className":40863,"style":16031},[88,16030],[19,40865],{"className":40866,"style":613},[411],[19,40868,29448],{"className":40869},[335],[19,40871],{"className":40872,"style":613},[411],[19,40874,40876,40879],{"className":40875},[66],[19,40877],{"className":40878,"style":1214},[70],[19,40880,39228],{"className":40881},[75],", the power series diverges. 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are:",[19,40943,40945],{"className":40944,"translate":23},[187],[19,40946,40948,41002],{"className":40947,"translate":23},[22],[19,40949,40951],{"className":40950},[27],[29,40952,40953],{"xmlns":31,"display":196},[33,40954,40955,40999],{},[36,40956,40957,40971],{},[26840,40958,40959,40961,40969],{},[43,40960,26844],{},[36,40962,40963,40965,40967],{},[39,40964,3297],{},[43,40966,267],{},[108,40968,3495],{},[39,40970,26855],{"mathvariant":275},[226,40972,40973,40983],{},[269,40974,40975,40977],{},[108,40976,39228],{},[36,40978,40979,40981],{},[108,40980,3518],{},[39,40982,3297],{},[36,40984,40985,40987,40989],{},[39,40986,3297],{},[43,40988,240],{},[269,40990,40991,40993],{},[108,40992,39228],{},[36,40994,40995,40997],{},[108,40996,3518],{},[39,40998,3297],{},[54,41000,41001],{"encoding":56},"\\sum_{n=1}^{\\infty}{\\frac{3^{2n}}{n-3^{2n}}}",[19,41003,41005],{"className":41004,"ariaHidden":62},[61],[19,41006,41008,41011,41078,41081],{"className":41007},[66],[19,41009],{"className":41010,"style":26984},[70],[19,41012,41014],{"className":41013},[304,305],[19,41015,41017,41070],{"className":41016},[137,138],[19,41018,41020,41067],{"className":41019},[142],[19,41021,41023,41043,41053],{"className":41022,"style":26997},[146],[19,41024,41025,41028],{"style":27000},[19,41026],{"className":41027,"style":27004},[154],[19,41029,41031],{"className":41030},[159,160,161,162],[19,41032,41034,41037,41040],{"className":41033},[75,162],[19,41035,3297],{"className":41036},[75,76,162],[19,41038,267],{"className":41039},[335,162],[19,41041,3495],{"className":41042},[75,162],[19,41044,41045,41048],{"style":27022},[19,41046],{"className":41047,"style":27004},[154],[19,41049,41050],{},[19,41051,26844],{"className":41052},[304,27031,27032],[19,41054,41055,41058],{"style":27035},[19,41056],{"className":41057,"style":27004},[154],[19,41059,41061],{"className":41060},[159,160,161,162],[19,41062,41064],{"className":41063},[75,162],[19,41065,26855],{"className":41066},[75,162],[19,41068,173],{"className":41069},[172],[19,41071,41073],{"className":41072},[142],[19,41074,41076],{"className":41075,"style":27057},[146],[19,41077],{},[19,41079],{"className":41080,"style":412},[411],[19,41082,41084],{"className":41083},[75],[19,41085,41087,41090,41219],{"className":41086},[75],[19,41088],{"className":41089},[81,422],[19,41091,41093],{"className":41092},[226],[19,41094,41096,41211],{"className":41095},[137,138],[19,41097,41099,41208],{"className":41098},[142],[19,41100,41102,41157,41165],{"className":41101,"style":39338},[146],[19,41103,41104,41107],{"style":438},[19,41105],{"className":41106,"style":322},[154],[19,41108,41110,41113,41116,41119,41122],{"className":41109},[75],[19,41111,3297],{"className":41112},[75,76],[19,41114],{"className":41115,"style":451},[411],[19,41117,240],{"className":41118},[455],[19,41120],{"className":41121,"style":451},[411],[19,41123,41125,41128],{"className":41124},[75],[19,41126,39228],{"className":41127},[75],[19,41129,41131],{"className":41130},[133],[19,41132,41134],{"className":41133},[137],[19,41135,41137],{"className":41136},[142],[19,41138,41140],{"className":41139,"style":39377},[146],[19,41141,41142,41145],{"style":14745},[19,41143],{"className":41144,"style":155},[154],[19,41146,41148],{"className":41147},[159,160,161,162],[19,41149,41151,41154],{"className":41150},[75,162],[19,41152,3518],{"className":41153},[75,162],[19,41155,3297],{"className":41156},[75,76,162],[19,41158,41159,41162],{"style":504},[19,41160],{"className":41161,"style":322},[154],[19,41163],{"className":41164,"style":512},[511],[19,41166,41167,41170],{"style":515},[19,41168],{"className":41169,"style":322},[154],[19,41171,41173],{"className":41172},[75],[19,41174,41176,41179],{"className":41175},[75],[19,41177,39228],{"className":41178},[75],[19,41180,41182],{"className":41181},[133],[19,41183,41185],{"className":41184},[137],[19,41186,41188],{"className":41187},[142],[19,41189,41191],{"className":41190,"style":19038},[146],[19,41192,41193,41196],{"style":1643},[19,41194],{"className":41195,"style":155},[154],[19,41197,41199],{"className":41198},[159,160,161,162],[19,41200,41202,41205],{"className":41201},[75,162],[19,41203,3518],{"className":41204},[75,162],[19,41206,3297],{"className":41207},[75,76,162],[19,41209,173],{"className":41210},[172],[19,41212,41214],{"className":41213},[142],[19,41215,41217],{"className":41216,"style":39455},[146],[19,41218],{},[19,41220],{"className":41221},[88,422],[15,41223,11523],{},[19,41225,41227],{"className":41226,"translate":23},[187],[19,41228,41230,41296],{"className":41229,"translate":23},[22],[19,41231,41233],{"className":41232},[27],[29,41234,41235],{"xmlns":31,"display":196},[33,41236,41237,41293],{},[36,41238,41239,41255,41283,41285,41287,41289,41291],{},[202,41240,41241,41247],{},[36,41242,41243,41245],{},[39,41244,208],{},[43,41246,211],{},[36,41248,41249,41251,41253],{},[39,41250,3297],{},[43,41252,218],{},[39,41254,26855],{"mathvariant":275},[226,41256,41257,41267],{},[269,41258,4125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\\rightarrow \\infty}{\\frac{3^{2n}}{n-3^{2n}}}=-1 \\neq 0",[19,41297,41299,41512,41566],{"className":41298,"ariaHidden":62},[61],[19,41300,41302,41306,41359,41362,41503,41506,41509],{"className":41301},[66],[19,41303],{"className":41304,"style":41305},[70],"height:2.2604em;vertical-align:-0.7693em;",[19,41307,41309],{"className":41308},[304,305],[19,41310,41312,41351],{"className":41311},[137,138],[19,41313,41315,41348],{"className":41314},[142],[19,41316,41318,41338],{"className":41317,"style":315},[146],[19,41319,41320,41323],{"style":318},[19,41321],{"className":41322,"style":322},[154],[19,41324,41326],{"className":41325},[159,160,161,162],[19,41327,41329,41332,41335],{"className":41328},[75,162],[19,41330,3297],{"className":41331},[75,76,162],[19,41333,218],{"className":41334},[335,162],[19,41336,26855],{"className":41337},[75,162],[19,41339,41340,41343],{"style":387},[19,41341],{"className":41342,"style":322},[154],[19,41344,41345],{},[19,41346,208],{"className":41347},[304],[19,41349,173],{"className":41350},[172],[19,41352,41354],{"className":41353},[142],[19,41355,41357],{"className":41356,"style":35166},[146],[19,41358],{},[19,41360],{"className":41361,"style":412},[411],[19,41363,41365],{"className":41364},[75],[19,41366,41368,41371,41500],{"className":41367},[75],[19,41369],{"className":41370},[81,422],[19,41372,41374],{"className":41373},[226],[19,41375,41377,41492],{"className":41376},[137,138],[19,41378,41380,41489],{"className":41379},[142],[19,41381,41383,41438,41446],{"className":41382,"style":39338},[146],[19,41384,41385,41388],{"style":438},[19,41386],{"className":41387,"style":322},[154],[19,41389,41391,41394,41397,41400,41403],{"className":41390},[75],[19,41392,3297],{"className":41393},[75,76],[19,41395],{"className":41396,"style":451},[411],[19,41398,240],{"className":41399},[455],[19,41401],{"className":41402,"style":451},[411],[19,41404,41406,41409],{"className":41405},[75],[19,41407,39228],{"className":41408},[75],[19,41410,41412],{"className":41411},[133],[19,41413,41415],{"className":41414},[137],[19,41416,41418],{"className":41417},[142],[19,41419,41421],{"className":41420,"style":39377},[146],[19,41422,41423,41426],{"style":14745},[19,41424],{"className":41425,"style":155},[154],[19,41427,41429],{"className":41428},[159,160,161,162],[19,41430,41432,41435],{"className":41431},[75,162],[19,41433,3518],{"className":41434},[75,162],[19,41436,3297],{"className":41437},[75,76,162],[19,41439,41440,41443],{"style":504},[19,41441],{"className":41442,"style":322},[154],[19,41444],{"className":41445,"style":512},[511],[19,41447,41448,41451],{"style":515},[19,41449],{"className":41450,"style":322},[154],[19,41452,41454],{"className":41453},[75],[19,41455,41457,41460],{"className":41456},[75],[19,41458,39228],{"className":41459},[75],[19,41461,41463],{"className":41462},[133],[19,41464,41466],{"className":41465},[137],[19,41467,41469],{"className":41468},[142],[19,41470,41472],{"className":41471,"style":19038},[146],[19,41473,41474,41477],{"style":1643},[19,41475],{"className":41476,"style":155},[154],[19,41478,41480],{"className":41479},[159,160,161,162],[19,41481,41483,41486],{"className":41482},[75,162],[19,41484,3518],{"className":41485},[75,162],[19,41487,3297],{"className":41488},[75,76,162],[19,41490,173],{"className":41491},[172],[19,41493,41495],{"className":41494},[142],[19,41496,41498],{"className":41497,"style":39455},[146],[19,41499],{},[19,41501],{"className":41502},[88,422],[19,41504],{"className":41505,"style":613},[411],[19,41507,267],{"className":41508},[335],[19,41510],{"className":41511,"style":613},[411],[19,41513,41515,41518,41521,41524,41527,41563],{"className":41514},[66],[19,41516],{"className":41517,"style":6202},[70],[19,41519,240],{"className":41520},[75],[19,41522,3495],{"className":41523},[75],[19,41525],{"className":41526,"style":613},[411],[19,41528,41530,41557,41560],{"className":41529},[335],[19,41531,41533],{"className":41532},[335],[19,41534,41536],{"className":41535},[75,29069],[19,41537,41539],{"className":41538},[29073],[19,41540,41542,41545,41554],{"className":41541},[29077],[19,41543],{"className":41544,"style":6202},[70],[19,41546,41548],{"className":41547},[29084],[19,41549,41551],{"className":41550},[75],[19,41552,29091],{"className":41553},[335],[19,41555],{"className":41556},[29095],[19,41558],{"className":41559},[411,29099],[19,41561,267],{"className":41562},[335],[19,41564],{"className":41565,"style":613},[411],[19,41567,41569,41572],{"className":41568},[66],[19,41570],{"className":41571,"style":1214},[70],[19,41573,110],{"className":41574},[75],[15,41576,41577],{},"the 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indicating that the original series has a convergence interval of ",[19,41861,41863,41887],{"className":41862,"translate":23},[22],[19,41864,41866],{"className":41865},[27],[29,41867,41868],{"xmlns":31},[33,41869,41870,41884],{},[36,41871,41872,41874,41876,41878,41880,41882],{},[43,41873,46],{"stretchy":45},[43,41875,240],{},[108,41877,39228],{},[43,41879,7830],{"separator":62},[108,41881,39228],{},[43,41883,52],{"stretchy":45},[54,41885,41886],{"encoding":56},"(-3,3)",[19,41888,41890],{"className":41889,"ariaHidden":62},[61],[19,41891,41893,41896,41899,41902,41905,41908,41911,41914],{"className":41892},[66],[19,41894],{"className":41895,"style":71},[70],[19,41897,46],{"className":41898},[81],[19,41900,240],{"className":41901},[75],[19,41903,39228],{"className":41904},[75],[19,41906,7830],{"className":41907},[8036],[19,41909],{"className":41910,"style":412},[411],[19,41912,39228],{"className":41913},[75],[19,41915,52],{"className":41916},[88],[15,41918,41919],{},"Power series, compared to general series of functions, exhibit two properties regarding uniform convergence:",[15,41921,41922,28757],{},[7600,41923,41924],{},"Property 1:",[19,41926,41928],{"className":41927,"translate":23},[187],[19,41929,41931,41970],{"className":41930,"translate":23},[22],[19,41932,41934],{"className":41933},[27],[29,41935,41936],{"xmlns":31,"display":196},[33,41937,41938,41968],{},[36,41939,41940,41954],{},[26840,41941,41942,41944,41952],{},[43,41943,26844],{},[36,41945,41946,41948,41950],{},[39,41947,3297],{},[43,41949,267],{},[108,41951,110],{},[39,41953,26855],{"mathvariant":275},[36,41955,41956,41962],{},[103,41957,41958,41960],{},[39,41959,3484],{},[39,41961,3297],{},[269,41963,41964,41966],{},[39,41965,49],{},[39,41967,3297],{},[54,41969,28591],{"encoding":56},[19,41971,41973],{"className":41972,"ariaHidden":62},[61],[19,41974,41976,41979,42046,42049],{"className":41975},[66],[19,41977],{"className":41978,"style":26984},[70],[19,41980,41982],{"className":41981},[304,305],[19,41983,41985,42038],{"className":41984},[137,138],[19,41986,41988,42035],{"className":41987},[142],[19,41989,41991,42011,42021],{"className":41990,"style":26997},[146],[19,41992,41993,41996],{"style":27000},[19,41994],{"className":41995,"style":27004},[154],[19,41997,41999],{"className":41998},[159,160,161,162],[19,42000,42002,42005,42008],{"className":42001},[75,162],[19,42003,3297],{"className":42004},[75,76,162],[19,42006,267],{"className":42007},[335,162],[19,42009,110],{"className":42010},[75,162],[19,42012,42013,42016],{"style":27022},[19,42014],{"className":42015,"style":27004},[154],[19,42017,42018],{},[19,42019,26844],{"className":42020},[304,27031,27032],[19,42022,42023,42026],{"style":27035},[19,42024],{"className":42025,"style":27004},[154],[19,42027,42029],{"className":42028},[159,160,161,162],[19,42030,42032],{"className":42031},[75,162],[19,42033,26855],{"className":42034},[75,162],[19,42036,173],{"className":42037},[172],[19,42039,42041],{"className":42040},[142],[19,42042,42044],{"className":42043,"style":27057},[146],[19,42045],{},[19,42047],{"className":42048,"style":412},[411],[19,42050,42052,42095],{"className":42051},[75],[19,42053,42055,42058],{"className":42054},[75],[19,42056,3484],{"className":42057},[75,76],[19,42059,42061],{"className":42060},[133],[19,42062,42064,42087],{"className":42063},[137,138],[19,42065,42067,42084],{"className":42066},[142],[19,42068,42070],{"className":42069,"style":3609},[146],[19,42071,42072,42075],{"style":150},[19,42073],{"className":42074,"style":155},[154],[19,42076,42078],{"className":42077},[159,160,161,162],[19,42079,42081],{"className":42080},[75,162],[19,42082,3297],{"className":42083},[75,76,162],[19,42085,173],{"className":42086},[172],[19,42088,42090],{"className":42089},[142],[19,42091,42093],{"className":42092,"style":180},[146],[19,42094],{},[19,42096,42098,42101],{"className":42097},[75],[19,42099,49],{"className":42100},[75,76],[19,42102,42104],{"className":42103},[133],[19,42105,42107],{"className":42106},[137],[19,42108,42110],{"className":42109},[142],[19,42111,42113],{"className":42112,"style":4148},[146],[19,42114,42115,42118],{"style":648},[19,42116],{"className":42117,"style":155},[154],[19,42119,42121],{"className":42120},[159,160,161,162],[19,42122,42124],{"className":42123},[75,162],[19,42125,3297],{"className":42126},[75,76,162],[15,42128,42129,42130,42188],{},"has a convergence radius ",[19,42131,42133,42155],{"className":42132,"translate":23},[22],[19,42134,42136],{"className":42135},[27],[29,42137,42138],{"xmlns":31},[33,42139,42140,42152],{},[36,42141,42142,42144,42146,42148,42150],{},[39,42143,10758],{},[43,42145,46],{"stretchy":45},[43,42147,29448],{},[108,42149,110],{},[43,42151,52],{"stretchy":45},[54,42153,42154],{"encoding":56},"R (> 0)",[19,42156,42158,42176],{"className":42157,"ariaHidden":62},[61],[19,42159,42161,42164,42167,42170,42173],{"className":42160},[66],[19,42162],{"className":42163,"style":71},[70],[19,42165,10758],{"className":42166,"style":10844},[75,76],[19,42168,46],{"className":42169},[81],[19,42171,29448],{"className":42172},[335],[19,42174],{"className":42175,"style":613},[411],[19,42177,42179,42182,42185],{"className":42178},[66],[19,42180],{"className":42181,"style":71},[70],[19,42183,110],{"className":42184},[75],[19,42186,52],{"className":42187},[88],", then the power series uniformly converges on its convergence interval ((-R,R)).",[15,42190,42191,42193,42194,42246,42247,42303],{},[7600,42192,10737],{}," Let ",[19,42195,42197,42219],{"className":42196,"translate":23},[22],[19,42198,42200],{"className":42199},[27],[29,42201,42202],{"xmlns":31},[33,42203,42204,42216],{},[36,42205,42206,42208,42210,42212,42214],{},[43,42207,14194],{"stretchy":45},[39,42209,3484],{},[43,42211,7830],{"separator":62},[39,42213,17223],{},[43,42215,14287],{"stretchy":45},[54,42217,42218],{"encoding":56},"[a,b]",[19,42220,42222],{"className":42221,"ariaHidden":62},[61],[19,42223,42225,42228,42231,42234,42237,42240,42243],{"className":42224},[66],[19,42226],{"className":42227,"style":71},[70],[19,42229,14194],{"className":42230},[81],[19,42232,3484],{"className":42233},[75,76],[19,42235,7830],{"className":42236},[8036],[19,42238],{"className":42239,"style":412},[411],[19,42241,17223],{"className":42242},[75,76],[19,42244,14287],{"className":42245},[88]," be any closed interval within ",[19,42248,42250,42273],{"className":42249,"translate":23},[22],[19,42251,42253],{"className":42252},[27],[29,42254,42255],{"xmlns":31},[33,42256,42257,42271],{},[36,42258,42259,42261,42263,42265,42267,42269],{},[43,42260,46],{"stretchy":45},[43,42262,240],{},[39,42264,10758],{},[43,42266,7830],{"separator":62},[39,42268,10758],{},[43,42270,52],{"stretchy":45},[54,42272,33497],{"encoding":56},[19,42274,42276],{"className":42275,"ariaHidden":62},[61],[19,42277,42279,42282,42285,42288,42291,42294,42297,42300],{"className":42278},[66],[19,42280],{"className":42281,"style":71},[70],[19,42283,46],{"className":42284},[81],[19,42286,240],{"className":42287},[75],[19,42289,10758],{"className":42290,"style":10844},[75,76],[19,42292,7830],{"className":42293},[8036],[19,42295],{"className":42296,"style":412},[411],[19,42298,10758],{"className":42299,"style":10844},[75,76],[19,42301,52],{"className":42302},[88],", and denote",[19,42305,42307],{"className":42306,"translate":23},[187],[19,42308,42310,42373],{"className":42309,"translate":23},[22],[19,42311,42313],{"className":42312},[27],[29,42314,42315],{"xmlns":31,"display":196},[33,42316,42317,42370],{},[36,42318,42319,42325,42327,42330,42332,42356,42358,42360,42362,42364,42366,42368],{},[28980,42320,42321,42323],{"accent":62},[39,42322,49],{},[43,42324,28986],{},[43,42326,267],{},[39,42328,42329],{},"max",[43,42331,211],{},[36,42333,42334,42336,42344,42346,42354],{},[43,42335,26641],{"fence":62},[36,42337,42338,42340,42342],{},[43,42339,29161],{"fence":62},[39,42341,3484],{},[43,42343,29161],{"fence":62},[43,42345,7830],{"separator":62},[36,42347,42348,42350,42352],{},[43,42349,29161],{"fence":62},[39,42351,17223],{},[43,42353,29161],{"fence":62},[43,42355,26668],{"fence":62},[43,42357,13533],{},[43,42359,46],{"stretchy":45},[43,42361,240],{},[39,42363,10758],{},[43,42365,7830],{"separator":62},[39,42367,10758],{},[43,42369,52],{"stretchy":45},[54,42371,42372],{"encoding":56},"\\bar{x} = \\max\\left\\{ \\left| a \\right|, \\left| b \\right| \\right\\} \\in (-R,R)",[19,42374,42376,42422,42485],{"className":42375,"ariaHidden":62},[61],[19,42377,42379,42382,42413,42416,42419],{"className":42378},[66],[19,42380],{"className":42381,"style":29035},[70],[19,42383,42385],{"className":42384},[75,29025],[19,42386,42388],{"className":42387},[137],[19,42389,42391],{"className":42390},[142],[19,42392,42394,42402],{"className":42393,"style":29035},[146],[19,42395,42396,42399],{"style":387},[19,42397],{"className":42398,"style":322},[154],[19,42400,49],{"className":42401},[75,76],[19,42403,42404,42407],{"style":387},[19,42405],{"className":42406,"style":322},[154],[19,42408,42410],{"className":42409,"style":29053},[29052],[19,42411,28986],{"className":42412},[75],[19,42414],{"className":42415,"style":613},[411],[19,42417,267],{"className":42418},[335],[19,42420],{"className":42421,"style":613},[411],[19,42423,42425,42428,42431,42434,42476,42479,42482],{"className":42424},[66],[19,42426],{"className":42427,"style":71},[70],[19,42429,42329],{"className":42430},[304],[19,42432],{"className":42433,"style":412},[411],[19,42435,42437,42440,42452,42455,42458,42461,42473],{"className":42436},[16026],[19,42438,26641],{"className":42439,"style":16031},[81,16030],[19,42441,42443,42446,42449],{"className":42442},[16026],[19,42444,29161],{"className":42445,"style":16031},[81,16030],[19,42447,3484],{"className":42448},[75,76],[19,42450,29161],{"className":42451,"style":16031},[88,16030],[19,42453],{"className":42454,"style":412},[411],[19,42456,7830],{"className":42457},[8036],[19,42459],{"className":42460,"style":412},[411],[19,42462,42464,42467,42470],{"className":42463},[16026],[19,42465,29161],{"className":42466,"style":16031},[81,16030],[19,42468,17223],{"className":42469},[75,76],[19,42471,29161],{"className":42472,"style":16031},[88,16030],[19,42474,26668],{"className":42475,"style":16031},[88,16030],[19,42477],{"className":42478,"style":613},[411],[19,42480,13533],{"className":42481},[335],[19,42483],{"className":42484,"style":613},[411],[19,42486,42488,42491,42494,42497,42500,42503,42506,42509],{"className":42487},[66],[19,42489],{"className":42490,"style":71},[70],[19,42492,46],{"className":42493},[81],[19,42495,240],{"className":42496},[75],[19,42498,10758],{"className":42499,"style":10844},[75,76],[19,42501,7830],{"className":42502},[8036],[19,42504],{"className":42505,"style":412},[411],[19,42507,10758],{"className":42508,"style":10844},[75,76],[19,42510,52],{"className":42511},[88],[15,42513,42514,42515,42543,42544,42595],{},"Then for any point ",[19,42516,42518,42531],{"className":42517,"translate":23},[22],[19,42519,42521],{"className":42520},[27],[29,42522,42523],{"xmlns":31},[33,42524,42525,42529],{},[36,42526,42527],{},[39,42528,49],{},[54,42530,49],{"encoding":56},[19,42532,42534],{"className":42533,"ariaHidden":62},[61],[19,42535,42537,42540],{"className":42536},[66],[19,42538],{"className":42539,"style":3309},[70],[19,42541,49],{"className":42542},[75,76]," in ",[19,42545,42547,42568],{"className":42546,"translate":23},[22],[19,42548,42550],{"className":42549},[27],[29,42551,42552],{"xmlns":31},[33,42553,42554,42566],{},[36,42555,42556,42558,42560,42562,42564],{},[43,42557,14194],{"stretchy":45},[39,42559,3484],{},[43,42561,7830],{"separator":62},[39,42563,17223],{},[43,42565,14287],{"stretchy":45},[54,42567,42218],{"encoding":56},[19,42569,42571],{"className":42570,"ariaHidden":62},[61],[19,42572,42574,42577,42580,42583,42586,42589,42592],{"className":42573},[66],[19,42575],{"className":42576,"style":71},[70],[19,42578,14194],{"className":42579},[81],[19,42581,3484],{"className":42582},[75,76],[19,42584,7830],{"className":42585},[8036],[19,42587],{"className":42588,"style":412},[411],[19,42590,17223],{"className":42591},[75,76],[19,42593,14287],{"className":42594},[88],", we have",[19,42597,42599],{"className":42598,"translate":23},[187],[19,42600,42602,42657],{"className":42601,"translate":23},[22],[19,42603,42605],{"className":42604},[27],[29,42606,42607],{"xmlns":31,"display":196},[33,42608,42609,42654],{},[36,42610,42611,42629,42632],{},[36,42612,42613,42615,42621,42627],{},[43,42614,29161],{"fence":62},[103,42616,42617,42619],{},[39,42618,3484],{},[39,42620,3297],{},[269,42622,42623,42625],{},[39,42624,49],{},[39,42626,3297],{},[43,42628,29161],{"fence":62},[43,42630,42631],{},"≤",[36,42633,42634,42636,42642,42652],{},[43,42635,29161],{"fence":62},[103,42637,42638,42640],{},[39,42639,3484],{},[39,42641,3297],{},[269,42643,42644,42650],{},[28980,42645,42646,42648],{"accent":62},[39,42647,49],{},[43,42649,28986],{},[39,42651,3297],{},[43,42653,29161],{"fence":62},[54,42655,42656],{"encoding":56},"\\left| a_{n}x^{n} \\right| \\leq \\left| a_{n}\\bar{x}^{n} \\right|",[19,42658,42660,42759],{"className":42659,"ariaHidden":62},[61],[19,42661,42663,42666,42750,42753,42756],{"className":42662},[66],[19,42664],{"className":42665,"style":71},[70],[19,42667,42669,42672,42715,42747],{"className":42668},[16026],[19,42670,29161],{"className":42671,"style":16031},[81,16030],[19,42673,42675,42678],{"className":42674},[75],[19,42676,3484],{"className":42677},[75,76],[19,42679,42681],{"className":42680},[133],[19,42682,42684,42707],{"className":42683},[137,138],[19,42685,42687,42704],{"className":42686},[142],[19,42688,42690],{"className":42689,"style":3609},[146],[19,42691,42692,42695],{"style":150},[19,42693],{"className":42694,"style":155},[154],[19,42696,42698],{"className":42697},[159,160,161,162],[19,42699,42701],{"className":42700},[75,162],[19,42702,3297],{"className":42703},[75,76,162],[19,42705,173],{"className":42706},[172],[19,42708,42710],{"className":42709},[142],[19,42711,42713],{"className":42712,"style":180},[146],[19,42714],{},[19,42716,42718,42721],{"className":42717},[75],[19,42719,49],{"className":42720},[75,76],[19,42722,42724],{"className":42723},[133],[19,42725,42727],{"className":42726},[137],[19,42728,42730],{"className":42729},[142],[19,42731,42733],{"className":42732,"style":4148},[146],[19,42734,42735,42738],{"style":648},[19,42736],{"className":42737,"style":155},[154],[19,42739,42741],{"className":42740},[159,160,161,162],[19,42742,42744],{"className":42743},[75,162],[19,42745,3297],{"className":42746},[75,76,162],[19,42748,29161],{"className":42749,"style":16031},[88,16030],[19,42751],{"className":42752,"style":613},[411],[19,42754,42631],{"className":42755},[335],[19,42757],{"className":42758,"style":613},[411],[19,42760,42762,42765],{"className":42761},[66],[19,42763],{"className":42764,"style":71},[70],[19,42766,42768,42771,42814,42874],{"className":42767},[16026],[19,42769,29161],{"className":42770,"style":16031},[81,16030],[19,42772,42774,42777],{"className":42773},[75],[19,42775,3484],{"className":42776},[75,76],[19,42778,42780],{"className":42779},[133],[19,42781,42783,42806],{"className":42782},[137,138],[19,42784,42786,42803],{"className":42785},[142],[19,42787,42789],{"className":42788,"style":3609},[146],[19,42790,42791,42794],{"style":150},[19,42792],{"className":42793,"style":155},[154],[19,42795,42797],{"className":42796},[159,160,161,162],[19,42798,42800],{"className":42799},[75,162],[19,42801,3297],{"className":42802},[75,76,162],[19,42804,173],{"className":42805},[172],[19,42807,42809],{"className":42808},[142],[19,42810,42812],{"className":42811,"style":180},[146],[19,42813],{},[19,42815,42817,42848],{"className":42816},[75],[19,42818,42820],{"className":42819},[75,29025],[19,42821,42823],{"className":42822},[137],[19,42824,42826],{"className":42825},[142],[19,42827,42829,42837],{"className":42828,"style":29035},[146],[19,42830,42831,42834],{"style":387},[19,42832],{"className":42833,"style":322},[154],[19,42835,49],{"className":42836},[75,76],[19,42838,42839,42842],{"style":387},[19,42840],{"className":42841,"style":322},[154],[19,42843,42845],{"className":42844,"style":29053},[29052],[19,42846,28986],{"className":42847},[75],[19,42849,42851],{"className":42850},[133],[19,42852,42854],{"className":42853},[137],[19,42855,42857],{"className":42856},[142],[19,42858,42860],{"className":42859,"style":4148},[146],[19,42861,42862,42865],{"style":648},[19,42863],{"className":42864,"style":155},[154],[19,42866,42868],{"className":42867},[159,160,161,162],[19,42869,42871],{"className":42870},[75,162],[19,42872,3297],{"className":42873},[75,76,162],[19,42875,29161],{"className":42876,"style":16031},[88,16030],[15,42878,42879,42880,42941],{},"Since the power series converges absolutely at the point ",[19,42881,42883,42901],{"className":42882,"translate":23},[22],[19,42884,42886],{"className":42885},[27],[29,42887,42888],{"xmlns":31},[33,42889,42890,42898],{},[36,42891,42892],{},[28980,42893,42894,42896],{"accent":62},[39,42895,49],{},[43,42897,28986],{},[54,42899,42900],{"encoding":56},"\\bar{x}",[19,42902,42904],{"className":42903,"ariaHidden":62},[61],[19,42905,42907,42910],{"className":42906},[66],[19,42908],{"className":42909,"style":29035},[70],[19,42911,42913],{"className":42912},[75,29025],[19,42914,42916],{"className":42915},[137],[19,42917,42919],{"className":42918},[142],[19,42920,42922,42930],{"className":42921,"style":29035},[146],[19,42923,42924,42927],{"style":387},[19,42925],{"className":42926,"style":322},[154],[19,42928,49],{"className":42929},[75,76],[19,42931,42932,42935],{"style":387},[19,42933],{"className":42934,"style":322},[154],[19,42936,42938],{"className":42937,"style":29053},[29052],[19,42939,28986],{"className":42940},[75],", by the M-test, the power series",[19,42943,42945],{"className":42944,"translate":23},[187],[19,42946,42948,42987],{"className":42947,"translate":23},[22],[19,42949,42951],{"className":42950},[27],[29,42952,42953],{"xmlns":31,"display":196},[33,42954,42955,42985],{},[36,42956,42957,42971],{},[26840,42958,42959,42961,42969],{},[43,42960,26844],{},[36,42962,42963,42965,42967],{},[39,42964,3297],{},[43,42966,267],{},[108,42968,110],{},[39,42970,26855],{"mathvariant":275},[36,42972,42973,42979],{},[103,42974,42975,42977],{},[39,42976,3484],{},[39,42978,3297],{},[269,42980,42981,42983],{},[39,42982,49],{},[39,42984,3297],{},[54,42986,28591],{"encoding":56},[19,42988,42990],{"className":42989,"ariaHidden":62},[61],[19,42991,42993,42996,43063,43066],{"className":42992},[66],[19,42994],{"className":42995,"style":26984},[70],[19,42997,42999],{"className":42998},[304,305],[19,43000,43002,43055],{"className":43001},[137,138],[19,43003,43005,43052],{"className":43004},[142],[19,43006,43008,43028,43038],{"className":43007,"style":26997},[146],[19,43009,43010,43013],{"style":27000},[19,43011],{"className":43012,"style":27004},[154],[19,43014,43016],{"className":43015},[159,160,161,162],[19,43017,43019,43022,43025],{"className":43018},[75,162],[19,43020,3297],{"className":43021},[75,76,162],[19,43023,267],{"className":43024},[335,162],[19,43026,110],{"className":43027},[75,162],[19,43029,43030,43033],{"style":27022},[19,43031],{"className":43032,"style":27004},[154],[19,43034,43035],{},[19,43036,26844],{"className":43037},[304,27031,27032],[19,43039,43040,43043],{"style":27035},[19,43041],{"className":43042,"style":27004},[154],[19,43044,43046],{"className":43045},[159,160,161,162],[19,43047,43049],{"className":43048},[75,162],[19,43050,26855],{"className":43051},[75,162],[19,43053,173],{"className":43054},[172],[19,43056,43058],{"className":43057},[142],[19,43059,43061],{"className":43060,"style":27057},[146],[19,43062],{},[19,43064],{"className":43065,"style":412},[411],[19,43067,43069,43112],{"className":43068},[75],[19,43070,43072,43075],{"className":43071},[75],[19,43073,3484],{"className":43074},[75,76],[19,43076,43078],{"className":43077},[133],[19,43079,43081,43104],{"className":43080},[137,138],[19,43082,43084,43101],{"className":43083},[142],[19,43085,43087],{"className":43086,"style":3609},[146],[19,43088,43089,43092],{"style":150},[19,43090],{"className":43091,"style":155},[154],[19,43093,43095],{"className":43094},[159,160,161,162],[19,43096,43098],{"className":43097},[75,162],[19,43099,3297],{"className":43100},[75,76,162],[19,43102,173],{"className":43103},[172],[19,43105,43107],{"className":43106},[142],[19,43108,43110],{"className":43109,"style":180},[146],[19,43111],{},[19,43113,43115,43118],{"className":43114},[75],[19,43116,49],{"className":43117},[75,76],[19,43119,43121],{"className":43120},[133],[19,43122,43124],{"className":43123},[137],[19,43125,43127],{"className":43126},[142],[19,43128,43130],{"className":43129,"style":4148},[146],[19,43131,43132,43135],{"style":648},[19,43133],{"className":43134,"style":155},[154],[19,43136,43138],{"className":43137},[159,160,161,162],[19,43139,43141],{"className":43140},[75,162],[19,43142,3297],{"className":43143},[75,76,162],[15,43145,43146,43147,43198,43199,43250,43251,43307],{},"uniformly converges on ",[19,43148,43150,43171],{"className":43149,"translate":23},[22],[19,43151,43153],{"className":43152},[27],[29,43154,43155],{"xmlns":31},[33,43156,43157,43169],{},[36,43158,43159,43161,43163,43165,43167],{},[43,43160,14194],{"stretchy":45},[39,43162,3484],{},[43,43164,7830],{"separator":62},[39,43166,17223],{},[43,43168,14287],{"stretchy":45},[54,43170,42218],{"encoding":56},[19,43172,43174],{"className":43173,"ariaHidden":62},[61],[19,43175,43177,43180,43183,43186,43189,43192,43195],{"className":43176},[66],[19,43178],{"className":43179,"style":71},[70],[19,43181,14194],{"className":43182},[81],[19,43184,3484],{"className":43185},[75,76],[19,43187,7830],{"className":43188},[8036],[19,43190],{"className":43191,"style":412},[411],[19,43193,17223],{"className":43194},[75,76],[19,43196,14287],{"className":43197},[88],". Since ",[19,43200,43202,43223],{"className":43201,"translate":23},[22],[19,43203,43205],{"className":43204},[27],[29,43206,43207],{"xmlns":31},[33,43208,43209,43221],{},[36,43210,43211,43213,43215,43217,43219],{},[43,43212,14194],{"stretchy":45},[39,43214,3484],{},[43,43216,7830],{"separator":62},[39,43218,17223],{},[43,43220,14287],{"stretchy":45},[54,43222,42218],{"encoding":56},[19,43224,43226],{"className":43225,"ariaHidden":62},[61],[19,43227,43229,43232,43235,43238,43241,43244,43247],{"className":43228},[66],[19,43230],{"className":43231,"style":71},[70],[19,43233,14194],{"className":43234},[81],[19,43236,3484],{"className":43237},[75,76],[19,43239,7830],{"className":43240},[8036],[19,43242],{"className":43243,"style":412},[411],[19,43245,17223],{"className":43246},[75,76],[19,43248,14287],{"className":43249},[88]," was chosen arbitrarily, the power series uniformly converges on the convergence interval ",[19,43252,43254,43277],{"className":43253,"translate":23},[22],[19,43255,43257],{"className":43256},[27],[29,43258,43259],{"xmlns":31},[33,43260,43261,43275],{},[36,43262,43263,43265,43267,43269,43271,43273],{},[43,43264,46],{"stretchy":45},[43,43266,240],{},[39,43268,10758],{},[43,43270,7830],{"separator":62},[39,43272,10758],{},[43,43274,52],{"stretchy":45},[54,43276,33497],{"encoding":56},[19,43278,43280],{"className":43279,"ariaHidden":62},[61],[19,43281,43283,43286,43289,43292,43295,43298,43301,43304],{"className":43282},[66],[19,43284],{"className":43285,"style":71},[70],[19,43287,46],{"className":43288},[81],[19,43290,240],{"className":43291},[75],[19,43293,10758],{"className":43294,"style":10844},[75,76],[19,43296,7830],{"className":43297},[8036],[19,43299],{"className":43300,"style":412},[411],[19,43302,10758],{"className":43303,"style":10844},[75,76],[19,43305,52],{"className":43306},[88]," .",[15,43309,43310,43313,43314,43509,43510,43567,43568,43619,43620,43677,43678,43873,43874,43619,43926,43983],{},[7600,43311,43312],{},"Property 2:"," If the power series ",[19,43315,43317,43356],{"className":43316,"translate":23},[22],[19,43318,43320],{"className":43319},[27],[29,43321,43322],{"xmlns":31},[33,43323,43324,43354],{},[36,43325,43326,43340],{},[4285,43327,43328,43330,43338],{},[43,43329,26844],{},[36,43331,43332,43334,43336],{},[39,43333,3297],{},[43,43335,267],{},[108,43337,110],{},[39,43339,26855],{"mathvariant":275},[36,43341,43342,43348],{},[103,43343,43344,43346],{},[39,43345,3484],{},[39,43347,3297],{},[269,43349,43350,43352],{},[39,43351,49],{},[39,43353,3297],{},[54,43355,28591],{"encoding":56},[19,43357,43359],{"className":43358,"ariaHidden":62},[61],[19,43360,43362,43365,43428,43431],{"className":43361},[66],[19,43363],{"className":43364,"style":31929},[70],[19,43366,43368,43371],{"className":43367},[304],[19,43369,26844],{"className":43370,"style":31937},[304,27031,31936],[19,43372,43374],{"className":43373},[133],[19,43375,43377,43420],{"className":43376},[137,138],[19,43378,43380,43417],{"className":43379},[142],[19,43381,43383,43403],{"className":43382,"style":31950},[146],[19,43384,43385,43388],{"style":31953},[19,43386],{"className":43387,"style":155},[154],[19,43389,43391],{"className":43390},[159,160,161,162],[19,43392,43394,43397,43400],{"className":43393},[75,162],[19,43395,3297],{"className":43396},[75,76,162],[19,43398,267],{"className":43399},[335,162],[19,43401,110],{"className":43402},[75,162],[19,43404,43405,43408],{"style":31974},[19,43406],{"className":43407,"style":155},[154],[19,43409,43411],{"className":43410},[159,160,161,162],[19,43412,43414],{"className":43413},[75,162],[19,43415,26855],{"className":43416},[75,162],[19,43418,173],{"className":43419},[172],[19,43421,43423],{"className":43422},[142],[19,43424,43426],{"className":43425,"style":31996},[146],[19,43427],{},[19,43429],{"className":43430,"style":412},[411],[19,43432,43434,43477],{"className":43433},[75],[19,43435,43437,43440],{"className":43436},[75],[19,43438,3484],{"className":43439},[75,76],[19,43441,43443],{"className":43442},[133],[19,43444,43446,43469],{"className":43445},[137,138],[19,43447,43449,43466],{"className":43448},[142],[19,43450,43452],{"className":43451,"style":3609},[146],[19,43453,43454,43457],{"style":150},[19,43455],{"className":43456,"style":155},[154],[19,43458,43460],{"className":43459},[159,160,161,162],[19,43461,43463],{"className":43462},[75,162],[19,43464,3297],{"className":43465},[75,76,162],[19,43467,173],{"className":43468},[172],[19,43470,43472],{"className":43471},[142],[19,43473,43475],{"className":43474,"style":180},[146],[19,43476],{},[19,43478,43480,43483],{"className":43479},[75],[19,43481,49],{"className":43482},[75,76],[19,43484,43486],{"className":43485},[133],[19,43487,43489],{"className":43488},[137],[19,43490,43492],{"className":43491},[142],[19,43493,43495],{"className":43494,"style":3429},[146],[19,43496,43497,43500],{"style":1643},[19,43498],{"className":43499,"style":155},[154],[19,43501,43503],{"className":43502},[159,160,161,162],[19,43504,43506],{"className":43505},[75,162],[19,43507,3297],{"className":43508},[75,76,162]," has a convergence radius ",[19,43511,43513,43534],{"className":43512,"translate":23},[22],[19,43514,43516],{"className":43515},[27],[29,43517,43518],{"xmlns":31},[33,43519,43520,43532],{},[36,43521,43522,43524,43526,43528,43530],{},[39,43523,10758],{},[43,43525,46],{"stretchy":45},[43,43527,29448],{},[108,43529,110],{},[43,43531,52],{"stretchy":45},[54,43533,42154],{"encoding":56},[19,43535,43537,43555],{"className":43536,"ariaHidden":62},[61],[19,43538,43540,43543,43546,43549,43552],{"className":43539},[66],[19,43541],{"className":43542,"style":71},[70],[19,43544,10758],{"className":43545,"style":10844},[75,76],[19,43547,46],{"className":43548},[81],[19,43550,29448],{"className":43551},[335],[19,43553],{"className":43554,"style":613},[411],[19,43556,43558,43561,43564],{"className":43557},[66],[19,43559],{"className":43560,"style":71},[70],[19,43562,110],{"className":43563},[75],[19,43565,52],{"className":43566},[88],", and converges at ",[19,43569,43571,43589],{"className":43570,"translate":23},[22],[19,43572,43574],{"className":43573},[27],[29,43575,43576],{"xmlns":31},[33,43577,43578,43586],{},[36,43579,43580,43582,43584],{},[39,43581,49],{},[43,43583,267],{},[39,43585,10758],{},[54,43587,43588],{"encoding":56},"x = R",[19,43590,43592,43610],{"className":43591,"ariaHidden":62},[61],[19,43593,43595,43598,43601,43604,43607],{"className":43594},[66],[19,43596],{"className":43597,"style":3309},[70],[19,43599,49],{"className":43600},[75,76],[19,43602],{"className":43603,"style":613},[411],[19,43605,267],{"className":43606},[335],[19,43608],{"className":43609,"style":613},[411],[19,43611,43613,43616],{"className":43612},[66],[19,43614],{"className":43615,"style":7738},[70],[19,43617,10758],{"className":43618,"style":10844},[75,76]," (or ",[19,43621,43623,43643],{"className":43622,"translate":23},[22],[19,43624,43626],{"className":43625},[27],[29,43627,43628],{"xmlns":31},[33,43629,43630,43640],{},[36,43631,43632,43634,43636,43638],{},[39,43633,49],{},[43,43635,267],{},[43,43637,240],{},[39,43639,10758],{},[54,43641,43642],{"encoding":56},"x = -R",[19,43644,43646,43664],{"className":43645,"ariaHidden":62},[61],[19,43647,43649,43652,43655,43658,43661],{"className":43648},[66],[19,43650],{"className":43651,"style":3309},[70],[19,43653,49],{"className":43654},[75,76],[19,43656],{"className":43657,"style":613},[411],[19,43659,267],{"className":43660},[335],[19,43662],{"className":43663,"style":613},[411],[19,43665,43667,43671,43674],{"className":43666},[66],[19,43668],{"className":43669,"style":43670},[70],"height:0.7667em;vertical-align:-0.0833em;",[19,43672,240],{"className":43673},[75],[19,43675,10758],{"className":43676,"style":10844},[75,76],"), then the power series ",[19,43679,43681,43720],{"className":43680,"translate":23},[22],[19,43682,43684],{"className":43683},[27],[29,43685,43686],{"xmlns":31},[33,43687,43688,43718],{},[36,43689,43690,43704],{},[4285,43691,43692,43694,43702],{},[43,43693,26844],{},[36,43695,43696,43698,43700],{},[39,43697,3297],{},[43,43699,267],{},[108,43701,110],{},[39,43703,26855],{"mathvariant":275},[36,43705,43706,43712],{},[103,43707,43708,43710],{},[39,43709,3484],{},[39,43711,3297],{},[269,43713,43714,43716],{},[39,43715,49],{},[39,43717,3297],{},[54,43719,28591],{"encoding":56},[19,43721,43723],{"className":43722,"ariaHidden":62},[61],[19,43724,43726,43729,43792,43795],{"className":43725},[66],[19,43727],{"className":43728,"style":31929},[70],[19,43730,43732,43735],{"className":43731},[304],[19,43733,26844],{"className":43734,"style":31937},[304,27031,31936],[19,43736,43738],{"className":43737},[133],[19,43739,43741,43784],{"className":43740},[137,138],[19,43742,43744,43781],{"className":43743},[142],[19,43745,43747,43767],{"className":43746,"style":31950},[146],[19,43748,43749,43752],{"style":31953},[19,43750],{"className":43751,"style":155},[154],[19,43753,43755],{"className":43754},[159,160,161,162],[19,43756,43758,43761,43764],{"className":43757},[75,162],[19,43759,3297],{"className":43760},[75,76,162],[19,43762,267],{"className":43763},[335,162],[19,43765,110],{"className":43766},[75,162],[19,43768,43769,43772],{"style":31974},[19,43770],{"className":43771,"style":155},[154],[19,43773,43775],{"className":43774},[159,160,161,162],[19,43776,43778],{"className":43777},[75,162],[19,43779,26855],{"className":43780},[75,162],[19,43782,173],{"className":43783},[172],[19,43785,43787],{"className":43786},[142],[19,43788,43790],{"className":43789,"style":31996},[146],[19,43791],{},[19,43793],{"className":43794,"style":412},[411],[19,43796,43798,43841],{"className":43797},[75],[19,43799,43801,43804],{"className":43800},[75],[19,43802,3484],{"className":43803},[75,76],[19,43805,43807],{"className":43806},[133],[19,43808,43810,43833],{"className":43809},[137,138],[19,43811,43813,43830],{"className":43812},[142],[19,43814,43816],{"className":43815,"style":3609},[146],[19,43817,43818,43821],{"style":150},[19,43819],{"className":43820,"style":155},[154],[19,43822,43824],{"className":43823},[159,160,161,162],[19,43825,43827],{"className":43826},[75,162],[19,43828,3297],{"className":43829},[75,76,162],[19,43831,173],{"className":43832},[172],[19,43834,43836],{"className":43835},[142],[19,43837,43839],{"className":43838,"style":180},[146],[19,43840],{},[19,43842,43844,43847],{"className":43843},[75],[19,43845,49],{"className":43846},[75,76],[19,43848,43850],{"className":43849},[133],[19,43851,43853],{"className":43852},[137],[19,43854,43856],{"className":43855},[142],[19,43857,43859],{"className":43858,"style":3429},[146],[19,43860,43861,43864],{"style":1643},[19,43862],{"className":43863,"style":155},[154],[19,43865,43867],{"className":43866},[159,160,161,162],[19,43868,43870],{"className":43869},[75,162],[19,43871,3297],{"className":43872},[75,76,162]," uniformly converges on ",[19,43875,43877,43899],{"className":43876,"translate":23},[22],[19,43878,43880],{"className":43879},[27],[29,43881,43882],{"xmlns":31},[33,43883,43884,43896],{},[36,43885,43886,43888,43890,43892,43894],{},[43,43887,14194],{"stretchy":45},[108,43889,110],{},[43,43891,7830],{"separator":62},[39,43893,10758],{},[43,43895,14287],{"stretchy":45},[54,43897,43898],{"encoding":56},"[0,R]",[19,43900,43902],{"className":43901,"ariaHidden":62},[61],[19,43903,43905,43908,43911,43914,43917,43920,43923],{"className":43904},[66],[19,43906],{"className":43907,"style":71},[70],[19,43909,14194],{"className":43910},[81],[19,43912,110],{"className":43913},[75],[19,43915,7830],{"className":43916},[8036],[19,43918],{"className":43919,"style":412},[411],[19,43921,10758],{"className":43922,"style":10844},[75,76],[19,43924,14287],{"className":43925},[88],[19,43927,43929,43953],{"className":43928,"translate":23},[22],[19,43930,43932],{"className":43931},[27],[29,43933,43934],{"xmlns":31},[33,43935,43936,43950],{},[36,43937,43938,43940,43942,43944,43946,43948],{},[43,43939,14194],{"stretchy":45},[43,43941,240],{},[39,43943,10758],{},[43,43945,7830],{"separator":62},[108,43947,110],{},[43,43949,14287],{"stretchy":45},[54,43951,43952],{"encoding":56},"[-R,0]",[19,43954,43956],{"className":43955,"ariaHidden":62},[61],[19,43957,43959,43962,43965,43968,43971,43974,43977,43980],{"className":43958},[66],[19,43960],{"className":43961,"style":71},[70],[19,43963,14194],{"className":43964},[81],[19,43966,240],{"className":43967},[75],[19,43969,10758],{"className":43970,"style":10844},[75,76],[19,43972,7830],{"className":43973},[8036],[19,43975],{"className":43976,"style":412},[411],[19,43978,110],{"className":43979},[75],[19,43981,14287],{"className":43982},[88],").",[15,43985,43986,43988,43989,44039,44040,44114,44115,3447],{},[7600,43987,10737],{}," Suppose the power series converges at ",[19,43990,43992,44009],{"className":43991,"translate":23},[22],[19,43993,43995],{"className":43994},[27],[29,43996,43997],{"xmlns":31},[33,43998,43999,44007],{},[36,44000,44001,44003,44005],{},[39,44002,49],{},[43,44004,267],{},[39,44006,10758],{},[54,44008,43588],{"encoding":56},[19,44010,44012,44030],{"className":44011,"ariaHidden":62},[61],[19,44013,44015,44018,44021,44024,44027],{"className":44014},[66],[19,44016],{"className":44017,"style":3309},[70],[19,44019,49],{"className":44020},[75,76],[19,44022],{"className":44023,"style":613},[411],[19,44025,267],{"className":44026},[335],[19,44028],{"className":44029,"style":613},[411],[19,44031,44033,44036],{"className":44032},[66],[19,44034],{"className":44035,"style":7738},[70],[19,44037,10758],{"className":44038,"style":10844},[75,76],". For ",[19,44041,44043,44069],{"className":44042,"translate":23},[22],[19,44044,44046],{"className":44045},[27],[29,44047,44048],{"xmlns":31},[33,44049,44050,44066],{},[36,44051,44052,44054,44056,44058,44060,44062,44064],{},[39,44053,49],{},[43,44055,13533],{},[43,44057,14194],{"stretchy":45},[108,44059,110],{},[43,44061,7830],{"separator":62},[39,44063,10758],{},[43,44065,14287],{"stretchy":45},[54,44067,44068],{"encoding":56},"x \\in [0,R]",[19,44070,44072,44090],{"className":44071,"ariaHidden":62},[61],[19,44073,44075,44078,44081,44084,44087],{"className":44074},[66],[19,44076],{"className":44077,"style":13562},[70],[19,44079,49],{"className":44080},[75,76],[19,44082],{"className":44083,"style":613},[411],[19,44085,13533],{"className":44086},[335],[19,44088],{"className":44089,"style":613},[411],[19,44091,44093,44096,44099,44102,44105,44108,44111],{"className":44092},[66],[19,44094],{"className":44095,"style":71},[70],[19,44097,14194],{"className":44098},[81],[19,44100,110],{"className":44101},[75],[19,44103,7830],{"className":44104},[8036],[19,44106],{"className":44107,"style":412},[411],[19,44109,10758],{"className":44110,"style":10844},[75,76],[19,44112,14287],{"className":44113},[88],", we have ",[19,44116,44118,44202],{"className":44117,"translate":23},[22],[19,44119,44121],{"className":44120},[27],[29,44122,44123],{"xmlns":31},[33,44124,44125,44199],{},[36,44126,44127,44141,44155,44157,44171],{},[4285,44128,44129,44131,44139],{},[43,44130,26844],{},[36,44132,44133,44135,44137],{},[39,44134,3297],{},[43,44136,267],{},[108,44138,110],{},[39,44140,26855],{"mathvariant":275},[36,44142,44143,44149],{},[103,44144,44145,44147],{},[39,44146,3484],{},[39,44148,3297],{},[269,44150,44151,44153],{},[39,44152,49],{},[39,44154,3297],{},[43,44156,267],{},[4285,44158,44159,44161,44169],{},[43,44160,26844],{},[36,44162,44163,44165,44167],{},[39,44164,3297],{},[43,44166,267],{},[108,44168,110],{},[39,44170,26855],{"mathvariant":275},[36,44172,44173,44179,44185,44187,44193],{},[103,44174,44175,44177],{},[39,44176,3484],{},[39,44178,3297],{},[269,44180,44181,44183],{},[39,44182,10758],{},[39,44184,3297],{},[43,44186,46],{"stretchy":45},[226,44188,44189,44191],{},[39,44190,49],{},[39,44192,10758],{},[269,44194,44195,44197],{},[43,44196,52],{"stretchy":45},[39,44198,3297],{},[54,44200,44201],{"encoding":56},"\\sum_{n=0}^{\\infty}{a_{n}x^{n}} = \\sum_{n=0}^{\\infty}{a_{n}R^{n}(\\frac{x}{R})^{n}}",[19,44203,44205,44364],{"className":44204,"ariaHidden":62},[61],[19,44206,44208,44211,44274,44277,44355,44358,44361],{"className":44207},[66],[19,44209],{"className":44210,"style":31929},[70],[19,44212,44214,44217],{"className":44213},[304],[19,44215,26844],{"className":44216,"style":31937},[304,27031,31936],[19,44218,44220],{"className":44219},[133],[19,44221,44223,44266],{"className":44222},[137,138],[19,44224,44226,44263],{"className":44225},[142],[19,44227,44229,44249],{"className":44228,"style":31950},[146],[19,44230,44231,44234],{"style":31953},[19,44232],{"className":44233,"style":155},[154],[19,44235,44237],{"className":44236},[159,160,161,162],[19,44238,44240,44243,44246],{"className":44239},[75,162],[19,44241,3297],{"className":44242},[75,76,162],[19,44244,267],{"className":44245},[335,162],[19,44247,110],{"className":44248},[75,162],[19,44250,44251,44254],{"style":31974},[19,44252],{"className":44253,"style":155},[154],[19,44255,44257],{"className":44256},[159,160,161,162],[19,44258,44260],{"className":44259},[75,162],[19,44261,26855],{"className":44262},[75,162],[19,44264,173],{"className":44265},[172],[19,44267,44269],{"className":44268},[142],[19,44270,44272],{"className":44271,"style":31996},[146],[19,44273],{},[19,44275],{"className":44276,"style":412},[411],[19,44278,44280,44323],{"className":44279},[75],[19,44281,44283,44286],{"className":44282},[75],[19,44284,3484],{"className":44285},[75,76],[19,44287,44289],{"className":44288},[133],[19,44290,44292,44315],{"className":44291},[137,138],[19,44293,44295,44312],{"className":44294},[142],[19,44296,44298],{"className":44297,"style":3609},[146],[19,44299,44300,44303],{"style":150},[19,44301],{"className":44302,"style":155},[154],[19,44304,44306],{"className":44305},[159,160,161,162],[19,44307,44309],{"className":44308},[75,162],[19,44310,3297],{"className":44311},[75,76,162],[19,44313,173],{"className":44314},[172],[19,44316,44318],{"className":44317},[142],[19,44319,44321],{"className":44320,"style":180},[146],[19,44322],{},[19,44324,44326,44329],{"className":44325},[75],[19,44327,49],{"className":44328},[75,76],[19,44330,44332],{"className":44331},[133],[19,44333,44335],{"className":44334},[137],[19,44336,44338],{"className":44337},[142],[19,44339,44341],{"className":44340,"style":3429},[146],[19,44342,44343,44346],{"style":1643},[19,44344],{"className":44345,"style":155},[154],[19,44347,44349],{"className":44348},[159,160,161,162],[19,44350,44352],{"className":44351},[75,162],[19,44353,3297],{"className":44354},[75,76,162],[19,44356],{"className":44357,"style":613},[411],[19,44359,267],{"className":44360},[335],[19,44362],{"className":44363,"style":613},[411],[19,44365,44367,44371,44434,44437],{"className":44366},[66],[19,44368],{"className":44369,"style":44370},[70],"height:1.1493em;vertical-align:-0.345em;",[19,44372,44374,44377],{"className":44373},[304],[19,44375,26844],{"className":44376,"style":31937},[304,27031,31936],[19,44378,44380],{"className":44379},[133],[19,44381,44383,44426],{"className":44382},[137,138],[19,44384,44386,44423],{"className":44385},[142],[19,44387,44389,44409],{"className":44388,"style":31950},[146],[19,44390,44391,44394],{"style":31953},[19,44392],{"className":44393,"style":155},[154],[19,44395,44397],{"className":44396},[159,160,161,162],[19,44398,44400,44403,44406],{"className":44399},[75,162],[19,44401,3297],{"className":44402},[75,76,162],[19,44404,267],{"className":44405},[335,162],[19,44407,110],{"className":44408},[75,162],[19,44410,44411,44414],{"style":31974},[19,44412],{"className":44413,"style":155},[154],[19,44415,44417],{"className":44416},[159,160,161,162],[19,44418,44420],{"className":44419},[75,162],[19,44421,26855],{"className":44422},[75,162],[19,44424,173],{"className":44425},[172],[19,44427,44429],{"className":44428},[142],[19,44430,44432],{"className":44431,"style":31996},[146],[19,44433],{},[19,44435],{"className":44436,"style":412},[411],[19,44438,44440,44483,44515,44518,44586],{"className":44439},[75],[19,44441,44443,44446],{"className":44442},[75],[19,44444,3484],{"className":44445},[75,76],[19,44447,44449],{"className":44448},[133],[19,44450,44452,44475],{"className":44451},[137,138],[19,44453,44455,44472],{"className":44454},[142],[19,44456,44458],{"className":44457,"style":3609},[146],[19,44459,44460,44463],{"style":150},[19,44461],{"className":44462,"style":155},[154],[19,44464,44466],{"className":44465},[159,160,161,162],[19,44467,44469],{"className":44468},[75,162],[19,44470,3297],{"className":44471},[75,76,162],[19,44473,173],{"className":44474},[172],[19,44476,44478],{"className":44477},[142],[19,44479,44481],{"className":44480,"style":180},[146],[19,44482],{},[19,44484,44486,44489],{"className":44485},[75],[19,44487,10758],{"className":44488,"style":10844},[75,76],[19,44490,44492],{"className":44491},[133],[19,44493,44495],{"className":44494},[137],[19,44496,44498],{"className":44497},[142],[19,44499,44501],{"className":44500,"style":3429},[146],[19,44502,44503,44506],{"style":1643},[19,44504],{"className":44505,"style":155},[154],[19,44507,44509],{"className":44508},[159,160,161,162],[19,44510,44512],{"className":44511},[75,162],[19,44513,3297],{"className":44514},[75,76,162],[19,44516,46],{"className":44517},[81],[19,44519,44521,44524,44583],{"className":44520},[75],[19,44522],{"className":44523},[81,422],[19,44525,44527],{"className":44526},[226],[19,44528,44530,44575],{"className":44529},[137,138],[19,44531,44533,44572],{"className":44532},[142],[19,44534,44536,44550,44558],{"className":44535,"style":30501},[146],[19,44537,44538,44541],{"style":30504},[19,44539],{"className":44540,"style":322},[154],[19,44542,44544],{"className":44543},[159,160,161,162],[19,44545,44547],{"className":44546},[75,162],[19,44548,10758],{"className":44549,"style":10844},[75,76,162],[19,44551,44552,44555],{"style":504},[19,44553],{"className":44554,"style":322},[154],[19,44556],{"className":44557,"style":512},[511],[19,44559,44560,44563],{"style":30556},[19,44561],{"className":44562,"style":322},[154],[19,44564,44566],{"className":44565},[159,160,161,162],[19,44567,44569],{"className":44568},[75,162],[19,44570,49],{"className":44571},[75,76,162],[19,44573,173],{"className":44574},[172],[19,44576,44578],{"className":44577},[142],[19,44579,44581],{"className":44580,"style":30578},[146],[19,44582],{},[19,44584],{"className":44585},[88,422],[19,44587,44589,44592],{"className":44588},[88],[19,44590,52],{"className":44591},[88],[19,44593,44595],{"className":44594},[133],[19,44596,44598],{"className":44597},[137],[19,44599,44601],{"className":44600},[142],[19,44602,44604],{"className":44603,"style":3429},[146],[19,44605,44606,44609],{"style":1643},[19,44607],{"className":44608,"style":155},[154],[19,44610,44612],{"className":44611},[159,160,161,162],[19,44613,44615],{"className":44614},[75,162],[19,44616,3297],{"className":44617},[75,76,162],[15,44619,44620],{},"Since the series",[19,44622,44624],{"className":44623,"translate":23},[187],[19,44625,44627,44667],{"className":44626,"translate":23},[22],[19,44628,44630],{"className":44629},[27],[29,44631,44632],{"xmlns":31,"display":196},[33,44633,44634,44664],{},[36,44635,44636,44650],{},[26840,44637,44638,44640,44648],{},[43,44639,26844],{},[36,44641,44642,44644,44646],{},[39,44643,3297],{},[43,44645,267],{},[108,44647,110],{},[39,44649,26855],{"mathvariant":275},[36,44651,44652,44658],{},[103,44653,44654,44656],{},[39,44655,3484],{},[39,44657,3297],{},[269,44659,44660,44662],{},[39,44661,10758],{},[39,44663,3297],{},[54,44665,44666],{"encoding":56},"\\sum_{n=0}^{\\infty}{a_{n}R^{n}}",[19,44668,44670],{"className":44669,"ariaHidden":62},[61],[19,44671,44673,44676,44743,44746],{"className":44672},[66],[19,44674],{"className":44675,"style":26984},[70],[19,44677,44679],{"className":44678},[304,305],[19,44680,44682,44735],{"className":44681},[137,138],[19,44683,44685,44732],{"className":44684},[142],[19,44686,44688,44708,44718],{"className":44687,"style":26997},[146],[19,44689,44690,44693],{"style":27000},[19,44691],{"className":44692,"style":27004},[154],[19,44694,44696],{"className":44695},[159,160,161,162],[19,44697,44699,44702,44705],{"className":44698},[75,162],[19,44700,3297],{"className":44701},[75,76,162],[19,44703,267],{"className":44704},[335,162],[19,44706,110],{"className":44707},[75,162],[19,44709,44710,44713],{"style":27022},[19,44711],{"className":44712,"style":27004},[154],[19,44714,44715],{},[19,44716,26844],{"className":44717},[304,27031,27032],[19,44719,44720,44723],{"style":27035},[19,44721],{"className":44722,"style":27004},[154],[19,44724,44726],{"className":44725},[159,160,161,162],[19,44727,44729],{"className":44728},[75,162],[19,44730,26855],{"className":44731},[75,162],[19,44733,173],{"className":44734},[172],[19,44736,44738],{"className":44737},[142],[19,44739,44741],{"className":44740,"style":27057},[146],[19,44742],{},[19,44744],{"className":44745,"style":412},[411],[19,44747,44749,44792],{"className":44748},[75],[19,44750,44752,44755],{"className":44751},[75],[19,44753,3484],{"className":44754},[75,76],[19,44756,44758],{"className":44757},[133],[19,44759,44761,44784],{"className":44760},[137,138],[19,44762,44764,44781],{"className":44763},[142],[19,44765,44767],{"className":44766,"style":3609},[146],[19,44768,44769,44772],{"style":150},[19,44770],{"className":44771,"style":155},[154],[19,44773,44775],{"className":44774},[159,160,161,162],[19,44776,44778],{"className":44777},[75,162],[19,44779,3297],{"className":44780},[75,76,162],[19,44782,173],{"className":44783},[172],[19,44785,44787],{"className":44786},[142],[19,44788,44790],{"className":44789,"style":180},[146],[19,44791],{},[19,44793,44795,44798],{"className":44794},[75],[19,44796,10758],{"className":44797,"style":10844},[75,76],[19,44799,44801],{"className":44800},[133],[19,44802,44804],{"className":44803},[137],[19,44805,44807],{"className":44806},[142],[19,44808,44810],{"className":44809,"style":4148},[146],[19,44811,44812,44815],{"style":648},[19,44813],{"className":44814,"style":155},[154],[19,44816,44818],{"className":44817},[159,160,161,162],[19,44819,44821],{"className":44820},[75,162],[19,44822,3297],{"className":44823},[75,76,162],[15,44825,44826],{},"converges, and the function sequence",[19,44828,44830],{"className":44829,"translate":23},[187],[19,44831,44833,44863],{"className":44832,"translate":23},[22],[19,44834,44836],{"className":44835},[27],[29,44837,44838],{"xmlns":31,"display":196},[33,44839,44840,44860],{},[36,44841,44842,44844,44846,44852,44858],{},[43,44843,26641],{"fence":62},[43,44845,46],{"stretchy":45},[226,44847,44848,44850],{},[39,44849,49],{},[39,44851,10758],{},[269,44853,44854,44856],{},[43,44855,52],{"stretchy":45},[39,44857,3297],{},[43,44859,26668],{"fence":62},[54,44861,44862],{"encoding":56},"\\left\\{ (\\frac{x}{R})^{n} \\right\\}",[19,44864,44866],{"className":44865,"ariaHidden":62},[61],[19,44867,44869,44873],{"className":44868},[66],[19,44870],{"className":44871,"style":44872},[70],"height:1.836em;vertical-align:-0.686em;",[19,44874,44876,44883,44886,44948,44980],{"className":44875},[16026],[19,44877,44879],{"className":44878,"style":16031},[81,16030],[19,44880,26641],{"className":44881},[16035,44882],"size2",[19,44884,46],{"className":44885},[81],[19,44887,44889,44892,44945],{"className":44888},[75],[19,44890],{"className":44891},[81,422],[19,44893,44895],{"className":44894},[226],[19,44896,44898,44937],{"className":44897},[137,138],[19,44899,44901,44934],{"className":44900},[142],[19,44902,44904,44915,44923],{"className":44903,"style":31431},[146],[19,44905,44906,44909],{"style":438},[19,44907],{"className":44908,"style":322},[154],[19,44910,44912],{"className":44911},[75],[19,44913,10758],{"className":44914,"style":10844},[75,76],[19,44916,44917,44920],{"style":504},[19,44918],{"className":44919,"style":322},[154],[19,44921],{"className":44922,"style":512},[511],[19,44924,44925,44928],{"style":515},[19,44926],{"className":44927,"style":322},[154],[19,44929,44931],{"className":44930},[75],[19,44932,49],{"className":44933},[75,76],[19,44935,173],{"className":44936},[172],[19,44938,44940],{"className":44939},[142],[19,44941,44943],{"className":44942,"style":5671},[146],[19,44944],{},[19,44946],{"className":44947},[88,422],[19,44949,44951,44954],{"className":44950},[88],[19,44952,52],{"className":44953},[88],[19,44955,44957],{"className":44956},[133],[19,44958,44960],{"className":44959},[137],[19,44961,44963],{"className":44962},[142],[19,44964,44966],{"className":44965,"style":4148},[146],[19,44967,44968,44971],{"style":648},[19,44969],{"className":44970,"style":155},[154],[19,44972,44974],{"className":44973},[159,160,161,162],[19,44975,44977],{"className":44976},[75,162],[19,44978,3297],{"className":44979},[75,76,162],[19,44981,44983],{"className":44982,"style":16031},[88,16030],[19,44984,26668],{"className":44985},[16035,44882],[15,44987,44988,44989,183],{},"is decreasing and uniformly bounded on ",[19,44990,44992,45013],{"className":44991,"translate":23},[22],[19,44993,44995],{"className":44994},[27],[29,44996,44997],{"xmlns":31},[33,44998,44999,45011],{},[36,45000,45001,45003,45005,45007,45009],{},[43,45002,14194],{"stretchy":45},[108,45004,110],{},[43,45006,7830],{"separator":62},[39,45008,10758],{},[43,45010,14287],{"stretchy":45},[54,45012,43898],{"encoding":56},[19,45014,45016],{"className":45015,"ariaHidden":62},[61],[19,45017,45019,45022,45025,45028,45031,45034,45037],{"className":45018},[66],[19,45020],{"className":45021,"style":71},[70],[19,45023,14194],{"className":45024},[81],[19,45026,110],{"className":45027},[75],[19,45029,7830],{"className":45030},[8036],[19,45032],{"className":45033,"style":412},[411],[19,45035,10758],{"className":45036,"style":10844},[75,76],[19,45038,14287],{"className":45039},[88],[19,45041,45043],{"className":45042,"translate":23},[187],[19,45044,45046,45121],{"className":45045,"translate":23},[22],[19,45047,45049],{"className":45048},[27],[29,45050,45051],{"xmlns":31,"display":196},[33,45052,45053,45118],{},[36,45054,45055,45057,45060,45066,45068,45070,45076,45082,45084,45086,45088,45090,45092,45094,45100,45106,45108,45110,45112,45114,45116],{},[108,45056,3495],{},[43,45058,45059],{},"≥",[226,45061,45062,45064],{},[39,45063,49],{},[39,45065,10758],{},[43,45067,45059],{},[43,45069,46],{"stretchy":45},[226,45071,45072,45074],{},[39,45073,49],{},[39,45075,10758],{},[269,45077,45078,45080],{},[43,45079,52],{"stretchy":45},[39,45081,3297],{},[43,45083,45059],{},[39,45085,3447],{"mathvariant":275},[39,45087,3447],{"mathvariant":275},[39,45089,3447],{"mathvariant":275},[43,45091,45059],{},[43,45093,46],{"stretchy":45},[226,45095,45096,45098],{},[39,45097,49],{},[39,45099,10758],{},[269,45101,45102,45104],{},[43,45103,52],{"stretchy":45},[39,45105,3297],{},[43,45107,45059],{},[39,45109,3447],{"mathvariant":275},[39,45111,3447],{"mathvariant":275},[39,45113,3447],{"mathvariant":275},[43,45115,45059],{},[108,45117,110],{},[54,45119,45120],{"encoding":56},"1 \\geq \\frac{x}{R} \\geq (\\frac{x}{R})^{n} \\geq ... \\geq (\\frac{x}{R})^{n} \\geq ... \\geq 0",[19,45122,45124,45143,45221,45333,45352,45464,45482],{"className":45123,"ariaHidden":62},[61],[19,45125,45127,45131,45134,45137,45140],{"className":45126},[66],[19,45128],{"className":45129,"style":45130},[70],"height:0.7804em;vertical-align:-0.136em;",[19,45132,3495],{"className":45133},[75],[19,45135],{"className":45136,"style":613},[411],[19,45138,45059],{"className":45139},[335],[19,45141],{"className":45142,"style":613},[411],[19,45144,45146,45150,45212,45215,45218],{"className":45145},[66],[19,45147],{"className":45148,"style":45149},[70],"height:1.7936em;vertical-align:-0.686em;",[19,45151,45153,45156,45209],{"className":45152},[75],[19,45154],{"className":45155},[81,422],[19,45157,45159],{"className":45158},[226],[19,45160,45162,45201],{"className":45161},[137,138],[19,45163,45165,45198],{"className":45164},[142],[19,45166,45168,45179,45187],{"className":45167,"style":31431},[146],[19,45169,45170,45173],{"style":438},[19,45171],{"className":45172,"style":322},[154],[19,45174,45176],{"className":45175},[75],[19,45177,10758],{"className":45178,"style":10844},[75,76],[19,45180,45181,45184],{"style":504},[19,45182],{"className":45183,"style":322},[154],[19,45185],{"className":45186,"style":512},[511],[19,45188,45189,45192],{"style":515},[19,45190],{"className":45191,"style":322},[154],[19,45193,45195],{"className":45194},[75],[19,45196,49],{"className":45197},[75,76],[19,45199,173],{"className":45200},[172],[19,45202,45204],{"className":45203},[142],[19,45205,45207],{"className":45206,"style":5671},[146],[19,45208],{},[19,45210],{"className":45211},[88,422],[19,45213],{"className":45214,"style":613},[411],[19,45216,45059],{"className":45217},[335],[19,45219],{"className":45220,"style":613},[411],[19,45222,45224,45227,45230,45292,45324,45327,45330],{"className":45223},[66],[19,45225],{"className":45226,"style":45149},[70],[19,45228,46],{"className":45229},[81],[19,45231,45233,45236,45289],{"className":45232},[75],[19,45234],{"className":45235},[81,422],[19,45237,45239],{"className":45238},[226],[19,45240,45242,45281],{"className":45241},[137,138],[19,45243,45245,45278],{"className":45244},[142],[19,45246,45248,45259,45267],{"className":45247,"style":31431},[146],[19,45249,45250,45253],{"style":438},[19,45251],{"className":45252,"style":322},[154],[19,45254,45256],{"className":45255},[75],[19,45257,10758],{"className":45258,"style":10844},[75,76],[19,45260,45261,45264],{"style":504},[19,45262],{"className":45263,"style":322},[154],[19,45265],{"className":45266,"style":512},[511],[19,45268,45269,45272],{"style":515},[19,45270],{"className":45271,"style":322},[154],[19,45273,45275],{"className":45274},[75],[19,45276,49],{"className":45277},[75,76],[19,45279,173],{"className":45280},[172],[19,45282,45284],{"className":45283},[142],[19,45285,45287],{"className":45286,"style":5671},[146],[19,45288],{},[19,45290],{"className":45291},[88,422],[19,45293,45295,45298],{"className":45294},[88],[19,45296,52],{"className":45297},[88],[19,45299,45301],{"className":45300},[133],[19,45302,45304],{"className":45303},[137],[19,45305,45307],{"className":45306},[142],[19,45308,45310],{"className":45309,"style":4148},[146],[19,45311,45312,45315],{"style":648},[19,45313],{"className":45314,"style":155},[154],[19,45316,45318],{"className":45317},[159,160,161,162],[19,45319,45321],{"className":45320},[75,162],[19,45322,3297],{"className":45323},[75,76,162],[19,45325],{"className":45326,"style":613},[411],[19,45328,45059],{"className":45329},[335],[19,45331],{"className":45332,"style":613},[411],[19,45334,45336,45340,45343,45346,45349],{"className":45335},[66],[19,45337],{"className":45338,"style":45339},[70],"height:0.7719em;vertical-align:-0.136em;",[19,45341,4007],{"className":45342},[75],[19,45344],{"className":45345,"style":613},[411],[19,45347,45059],{"className":45348},[335],[19,45350],{"className":45351,"style":613},[411],[19,45353,45355,45358,45361,45423,45455,45458,45461],{"className":45354},[66],[19,45356],{"className":45357,"style":45149},[70],[19,45359,46],{"className":45360},[81],[19,45362,45364,45367,45420],{"className":45363},[75],[19,45365],{"className":45366},[81,422],[19,45368,45370],{"className":45369},[226],[19,45371,45373,45412],{"className":45372},[137,138],[19,45374,45376,45409],{"className":45375},[142],[19,45377,45379,45390,45398],{"className":45378,"style":31431},[146],[19,45380,45381,45384],{"style":438},[19,45382],{"className":45383,"style":322},[154],[19,45385,45387],{"className":45386},[75],[19,45388,10758],{"className":45389,"style":10844},[75,76],[19,45391,45392,45395],{"style":504},[19,45393],{"className":45394,"style":322},[154],[19,45396],{"className":45397,"style":512},[511],[19,45399,45400,45403],{"style":515},[19,45401],{"className":45402,"style":322},[154],[19,45404,45406],{"className":45405},[75],[19,45407,49],{"className":45408},[75,76],[19,45410,173],{"className":45411},[172],[19,45413,45415],{"className":45414},[142],[19,45416,45418],{"className":45417,"style":5671},[146],[19,45419],{},[19,45421],{"className":45422},[88,422],[19,45424,45426,45429],{"className":45425},[88],[19,45427,52],{"className":45428},[88],[19,45430,45432],{"className":45431},[133],[19,45433,45435],{"className":45434},[137],[19,45436,45438],{"className":45437},[142],[19,45439,45441],{"className":45440,"style":4148},[146],[19,45442,45443,45446],{"style":648},[19,45444],{"className":45445,"style":155},[154],[19,45447,45449],{"className":45448},[159,160,161,162],[19,45450,45452],{"className":45451},[75,162],[19,45453,3297],{"className":45454},[75,76,162],[19,45456],{"className":45457,"style":613},[411],[19,45459,45059],{"className":45460},[335],[19,45462],{"className":45463,"style":613},[411],[19,45465,45467,45470,45473,45476,45479],{"className":45466},[66],[19,45468],{"className":45469,"style":45339},[70],[19,45471,4007],{"className":45472},[75],[19,45474],{"className":45475,"style":613},[411],[19,45477,45059],{"className":45478},[335],[19,45480],{"className":45481,"style":613},[411],[19,45483,45485,45488],{"className":45484},[66],[19,45486],{"className":45487,"style":1214},[70],[19,45489,110],{"className":45490},[75],[15,45492,45493],{},"by Abel's test, the series",[19,45495,45497],{"className":45496,"translate":23},[187],[19,45498,45500,45539],{"className":45499,"translate":23},[22],[19,45501,45503],{"className":45502},[27],[29,45504,45505],{"xmlns":31,"display":196},[33,45506,45507,45537],{},[36,45508,45509,45523],{},[26840,45510,45511,45513,45521],{},[43,45512,26844],{},[36,45514,45515,45517,45519],{},[39,45516,3297],{},[43,45518,267],{},[108,45520,110],{},[39,45522,26855],{"mathvariant":275},[36,45524,45525,45531],{},[103,45526,45527,45529],{},[39,45528,3484],{},[39,45530,3297],{},[269,45532,45533,45535],{},[39,45534,49],{},[39,45536,3297],{},[54,45538,28591],{"encoding":56},[19,45540,45542],{"className":45541,"ariaHidden":62},[61],[19,45543,45545,45548,45615,45618],{"className":45544},[66],[19,45546],{"className":45547,"style":26984},[70],[19,45549,45551],{"className":45550},[304,305],[19,45552,45554,45607],{"className":45553},[137,138],[19,45555,45557,45604],{"className":45556},[142],[19,45558,45560,45580,45590],{"className":45559,"style":26997},[146],[19,45561,45562,45565],{"style":27000},[19,45563],{"className":45564,"style":27004},[154],[19,45566,45568],{"className":45567},[159,160,161,162],[19,45569,45571,45574,45577],{"className":45570},[75,162],[19,45572,3297],{"className":45573},[75,76,162],[19,45575,267],{"className":45576},[335,162],[19,45578,110],{"className":45579},[75,162],[19,45581,45582,45585],{"style":27022},[19,45583],{"className":45584,"style":27004},[154],[19,45586,45587],{},[19,45588,26844],{"className":45589},[304,27031,27032],[19,45591,45592,45595],{"style":27035},[19,45593],{"className":45594,"style":27004},[154],[19,45596,45598],{"className":45597},[159,160,161,162],[19,45599,45601],{"className":45600},[75,162],[19,45602,26855],{"className":45603},[75,162],[19,45605,173],{"className":45606},[172],[19,45608,45610],{"className":45609},[142],[19,45611,45613],{"className":45612,"style":27057},[146],[19,45614],{},[19,45616],{"className":45617,"style":412},[411],[19,45619,45621,45664],{"className":45620},[75],[19,45622,45624,45627],{"className":45623},[75],[19,45625,3484],{"className":45626},[75,76],[19,45628,45630],{"className":45629},[133],[19,45631,45633,45656],{"className":45632},[137,138],[19,45634,45636,45653],{"className":45635},[142],[19,45637,45639],{"className":45638,"style":3609},[146],[19,45640,45641,45644],{"style":150},[19,45642],{"className":45643,"style":155},[154],[19,45645,45647],{"className":45646},[159,160,161,162],[19,45648,45650],{"className":45649},[75,162],[19,45651,3297],{"className":45652},[75,76,162],[19,45654,173],{"className":45655},[172],[19,45657,45659],{"className":45658},[142],[19,45660,45662],{"className":45661,"style":180},[146],[19,45663],{},[19,45665,45667,45670],{"className":45666},[75],[19,45668,49],{"className":45669},[75,76],[19,45671,45673],{"className":45672},[133],[19,45674,45676],{"className":45675},[137],[19,45677,45679],{"className":45678},[142],[19,45680,45682],{"className":45681,"style":4148},[146],[19,45683,45684,45687],{"style":648},[19,45685],{"className":45686,"style":155},[154],[19,45688,45690],{"className":45689},[159,160,161,162],[19,45691,45693],{"className":45692},[75,162],[19,45694,3297],{"className":45695},[75,76,162],[15,45697,43146,45698,3447],{},[19,45699,45701,45722],{"className":45700,"translate":23},[22],[19,45702,45704],{"className":45703},[27],[29,45705,45706],{"xmlns":31},[33,45707,45708,45720],{},[36,45709,45710,45712,45714,45716,45718],{},[43,45711,14194],{"stretchy":45},[108,45713,110],{},[43,45715,7830],{"separator":62},[39,45717,10758],{},[43,45719,14287],{"stretchy":45},[54,45721,43898],{"encoding":56},[19,45723,45725],{"className":45724,"ariaHidden":62},[61],[19,45726,45728,45731,45734,45737,45740,45743,45746],{"className":45727},[66],[19,45729],{"className":45730,"style":71},[70],[19,45732,14194],{"className":45733},[81],[19,45735,110],{"className":45736},[75],[19,45738,7830],{"className":45739},[8036],[19,45741],{"className":45742,"style":412},[411],[19,45744,10758],{"className":45745,"style":10844},[75,76],[19,45747,14287],{"className":45748},[88],[15,45750,45751,45753,45754,45949,45950,46007,46008,46203,46204,46254,46255,3447],{},[7600,45752,16911],{}," In fact, since the power series ",[19,45755,45757,45796],{"className":45756,"translate":23},[22],[19,45758,45760],{"className":45759},[27],[29,45761,45762],{"xmlns":31},[33,45763,45764,45794],{},[36,45765,45766,45780],{},[4285,45767,45768,45770,45778],{},[43,45769,26844],{},[36,45771,45772,45774,45776],{},[39,45773,3297],{},[43,45775,267],{},[108,45777,110],{},[39,45779,26855],{"mathvariant":275},[36,45781,45782,45788],{},[103,45783,45784,45786],{},[39,45785,3484],{},[39,45787,3297],{},[269,45789,45790,45792],{},[39,45791,49],{},[39,45793,3297],{},[54,45795,28591],{"encoding":56},[19,45797,45799],{"className":45798,"ariaHidden":62},[61],[19,45800,45802,45805,45868,45871],{"className":45801},[66],[19,45803],{"className":45804,"style":31929},[70],[19,45806,45808,45811],{"className":45807},[304],[19,45809,26844],{"className":45810,"style":31937},[304,27031,31936],[19,45812,45814],{"className":45813},[133],[19,45815,45817,45860],{"className":45816},[137,138],[19,45818,45820,45857],{"className":45819},[142],[19,45821,45823,45843],{"className":45822,"style":31950},[146],[19,45824,45825,45828],{"style":31953},[19,45826],{"className":45827,"style":155},[154],[19,45829,45831],{"className":45830},[159,160,161,162],[19,45832,45834,45837,45840],{"className":45833},[75,162],[19,45835,3297],{"className":45836},[75,76,162],[19,45838,267],{"className":45839},[335,162],[19,45841,110],{"className":45842},[75,162],[19,45844,45845,45848],{"style":31974},[19,45846],{"className":45847,"style":155},[154],[19,45849,45851],{"className":45850},[159,160,161,162],[19,45852,45854],{"className":45853},[75,162],[19,45855,26855],{"className":45856},[75,162],[19,45858,173],{"className":45859},[172],[19,45861,45863],{"className":45862},[142],[19,45864,45866],{"className":45865,"style":31996},[146],[19,45867],{},[19,45869],{"className":45870,"style":412},[411],[19,45872,45874,45917],{"className":45873},[75],[19,45875,45877,45880],{"className":45876},[75],[19,45878,3484],{"className":45879},[75,76],[19,45881,45883],{"className":45882},[133],[19,45884,45886,45909],{"className":45885},[137,138],[19,45887,45889,45906],{"className":45888},[142],[19,45890,45892],{"className":45891,"style":3609},[146],[19,45893,45894,45897],{"style":150},[19,45895],{"className":45896,"style":155},[154],[19,45898,45900],{"className":45899},[159,160,161,162],[19,45901,45903],{"className":45902},[75,162],[19,45904,3297],{"className":45905},[75,76,162],[19,45907,173],{"className":45908},[172],[19,45910,45912],{"className":45911},[142],[19,45913,45915],{"className":45914,"style":180},[146],[19,45916],{},[19,45918,45920,45923],{"className":45919},[75],[19,45921,49],{"className":45922},[75,76],[19,45924,45926],{"className":45925},[133],[19,45927,45929],{"className":45928},[137],[19,45930,45932],{"className":45931},[142],[19,45933,45935],{"className":45934,"style":3429},[146],[19,45936,45937,45940],{"style":1643},[19,45938],{"className":45939,"style":155},[154],[19,45941,45943],{"className":45942},[159,160,161,162],[19,45944,45946],{"className":45945},[75,162],[19,45947,3297],{"className":45948},[75,76,162]," is uniformly convergent on ",[19,45951,45953,45977],{"className":45952,"translate":23},[22],[19,45954,45956],{"className":45955},[27],[29,45957,45958],{"xmlns":31},[33,45959,45960,45974],{},[36,45961,45962,45964,45966,45968,45970,45972],{},[43,45963,46],{"stretchy":45},[43,45965,240],{},[39,45967,10758],{},[43,45969,7830],{"separator":62},[108,45971,110],{},[43,45973,14287],{"stretchy":45},[54,45975,45976],{"encoding":56},"(-R,0]",[19,45978,45980],{"className":45979,"ariaHidden":62},[61],[19,45981,45983,45986,45989,45992,45995,45998,46001,46004],{"className":45982},[66],[19,45984],{"className":45985,"style":71},[70],[19,45987,46],{"className":45988},[81],[19,45990,240],{"className":45991},[75],[19,45993,10758],{"className":45994,"style":10844},[75,76],[19,45996,7830],{"className":45997},[8036],[19,45999],{"className":46000,"style":412},[411],[19,46002,110],{"className":46003},[75],[19,46005,14287],{"className":46006},[88]," , the conclusion can also be drawn: when the power series ",[19,46009,46011,46050],{"className":46010,"translate":23},[22],[19,46012,46014],{"className":46013},[27],[29,46015,46016],{"xmlns":31},[33,46017,46018,46048],{},[36,46019,46020,46034],{},[4285,46021,46022,46024,46032],{},[43,46023,26844],{},[36,46025,46026,46028,46030],{},[39,46027,3297],{},[43,46029,267],{},[108,46031,110],{},[39,46033,26855],{"mathvariant":275},[36,46035,46036,46042],{},[103,46037,46038,46040],{},[39,46039,3484],{},[39,46041,3297],{},[269,46043,46044,46046],{},[39,46045,49],{},[39,46047,3297],{},[54,46049,28591],{"encoding":56},[19,46051,46053],{"className":46052,"ariaHidden":62},[61],[19,46054,46056,46059,46122,46125],{"className":46055},[66],[19,46057],{"className":46058,"style":31929},[70],[19,46060,46062,46065],{"className":46061},[304],[19,46063,26844],{"className":46064,"style":31937},[304,27031,31936],[19,46066,46068],{"className":46067},[133],[19,46069,46071,46114],{"className":46070},[137,138],[19,46072,46074,46111],{"className":46073},[142],[19,46075,46077,46097],{"className":46076,"style":31950},[146],[19,46078,46079,46082],{"style":31953},[19,46080],{"className":46081,"style":155},[154],[19,46083,46085],{"className":46084},[159,160,161,162],[19,46086,46088,46091,46094],{"className":46087},[75,162],[19,46089,3297],{"className":46090},[75,76,162],[19,46092,267],{"className":46093},[335,162],[19,46095,110],{"className":46096},[75,162],[19,46098,46099,46102],{"style":31974},[19,46100],{"className":46101,"style":155},[154],[19,46103,46105],{"className":46104},[159,160,161,162],[19,46106,46108],{"className":46107},[75,162],[19,46109,26855],{"className":46110},[75,162],[19,46112,173],{"className":46113},[172],[19,46115,46117],{"className":46116},[142],[19,46118,46120],{"className":46119,"style":31996},[146],[19,46121],{},[19,46123],{"className":46124,"style":412},[411],[19,46126,46128,46171],{"className":46127},[75],[19,46129,46131,46134],{"className":46130},[75],[19,46132,3484],{"className":46133},[75,76],[19,46135,46137],{"className":46136},[133],[19,46138,46140,46163],{"className":46139},[137,138],[19,46141,46143,46160],{"className":46142},[142],[19,46144,46146],{"className":46145,"style":3609},[146],[19,46147,46148,46151],{"style":150},[19,46149],{"className":46150,"style":155},[154],[19,46152,46154],{"className":46153},[159,160,161,162],[19,46155,46157],{"className":46156},[75,162],[19,46158,3297],{"className":46159},[75,76,162],[19,46161,173],{"className":46162},[172],[19,46164,46166],{"className":46165},[142],[19,46167,46169],{"className":46168,"style":180},[146],[19,46170],{},[19,46172,46174,46177],{"className":46173},[75],[19,46175,49],{"className":46176},[75,76],[19,46178,46180],{"className":46179},[133],[19,46181,46183],{"className":46182},[137],[19,46184,46186],{"className":46185},[142],[19,46187,46189],{"className":46188,"style":3429},[146],[19,46190,46191,46194],{"style":1643},[19,46192],{"className":46193,"style":155},[154],[19,46195,46197],{"className":46196},[159,160,161,162],[19,46198,46200],{"className":46199},[75,162],[19,46201,3297],{"className":46202},[75,76,162]," converges at ",[19,46205,46207,46224],{"className":46206,"translate":23},[22],[19,46208,46210],{"className":46209},[27],[29,46211,46212],{"xmlns":31},[33,46213,46214,46222],{},[36,46215,46216,46218,46220],{},[39,46217,49],{},[43,46219,267],{},[39,46221,10758],{},[54,46223,43588],{"encoding":56},[19,46225,46227,46245],{"className":46226,"ariaHidden":62},[61],[19,46228,46230,46233,46236,46239,46242],{"className":46229},[66],[19,46231],{"className":46232,"style":3309},[70],[19,46234,49],{"className":46235},[75,76],[19,46237],{"className":46238,"style":613},[411],[19,46240,267],{"className":46241},[335],[19,46243],{"className":46244,"style":613},[411],[19,46246,46248,46251],{"className":46247},[66],[19,46249],{"className":46250,"style":7738},[70],[19,46252,10758],{"className":46253,"style":10844},[75,76],", the power series uniformly converges on ",[19,46256,46258,46282],{"className":46257,"translate":23},[22],[19,46259,46261],{"className":46260},[27],[29,46262,46263],{"xmlns":31},[33,46264,46265,46279],{},[36,46266,46267,46269,46271,46273,46275,46277],{},[43,46268,46],{"stretchy":45},[43,46270,240],{},[39,46272,10758],{},[43,46274,7830],{"separator":62},[39,46276,10758],{},[43,46278,14287],{"stretchy":45},[54,46280,46281],{"encoding":56},"(-R,R]",[19,46283,46285],{"className":46284,"ariaHidden":62},[61],[19,46286,46288,46291,46294,46297,46300,46303,46306,46309],{"className":46287},[66],[19,46289],{"className":46290,"style":71},[70],[19,46292,46],{"className":46293},[81],[19,46295,240],{"className":46296},[75],[19,46298,10758],{"className":46299,"style":10844},[75,76],[19,46301,7830],{"className":46302},[8036],[19,46304],{"className":46305,"style":412},[411],[19,46307,10758],{"className":46308,"style":10844},[75,76],[19,46310,14287],{"className":46311},[88],[15,46313,46314],{},"From the two properties regarding uniform convergence of power series, we can deduce other properties that power series possess:",[15,46316,46317,46355,46356,46551],{},[19,46318,46320,46337],{"className":46319,"translate":23},[22],[19,46321,46323],{"className":46322},[27],[29,46324,46325],{"xmlns":31},[33,46326,46327,46335],{},[36,46328,46329,46331,46333],{},[43,46330,46],{"stretchy":45},[39,46332,34238],{},[43,46334,52],{"stretchy":45},[54,46336,34243],{"encoding":56},[19,46338,46340],{"className":46339,"ariaHidden":62},[61],[19,46341,46343,46346,46349,46352],{"className":46342},[66],[19,46344],{"className":46345,"style":71},[70],[19,46347,46],{"className":46348},[81],[19,46350,34238],{"className":46351},[75,76],[19,46353,52],{"className":46354},[88]," The sum function of the power series ",[19,46357,46359,46398],{"className":46358,"translate":23},[22],[19,46360,46362],{"className":46361},[27],[29,46363,46364],{"xmlns":31},[33,46365,46366,46396],{},[36,46367,46368,46382],{},[4285,46369,46370,46372,46380],{},[43,46371,26844],{},[36,46373,46374,46376,46378],{},[39,46375,3297],{},[43,46377,267],{},[108,46379,110],{},[39,46381,26855],{"mathvariant":275},[36,46383,46384,46390],{},[103,46385,46386,46388],{},[39,46387,3484],{},[39,46389,3297],{},[269,46391,46392,46394],{},[39,46393,49],{},[39,46395,3297],{},[54,46397,28591],{"encoding":56},[19,46399,46401],{"className":46400,"ariaHidden":62},[61],[19,46402,46404,46407,46470,46473],{"className":46403},[66],[19,46405],{"className":46406,"style":31929},[70],[19,46408,46410,46413],{"className":46409},[304],[19,46411,26844],{"className":46412,"style":31937},[304,27031,31936],[19,46414,46416],{"className":46415},[133],[19,46417,46419,46462],{"className":46418},[137,138],[19,46420,46422,46459],{"className":46421},[142],[19,46423,46425,46445],{"className":46424,"style":31950},[146],[19,46426,46427,46430],{"style":31953},[19,46428],{"className":46429,"style":155},[154],[19,46431,46433],{"className":46432},[159,160,161,162],[19,46434,46436,46439,46442],{"className":46435},[75,162],[19,46437,3297],{"className":46438},[75,76,162],[19,46440,267],{"className":46441},[335,162],[19,46443,110],{"className":46444},[75,162],[19,46446,46447,46450],{"style":31974},[19,46448],{"className":46449,"style":155},[154],[19,46451,46453],{"className":46452},[159,160,161,162],[19,46454,46456],{"className":46455},[75,162],[19,46457,26855],{"className":46458},[75,162],[19,46460,173],{"className":46461},[172],[19,46463,46465],{"className":46464},[142],[19,46466,46468],{"className":46467,"style":31996},[146],[19,46469],{},[19,46471],{"className":46472,"style":412},[411],[19,46474,46476,46519],{"className":46475},[75],[19,46477,46479,46482],{"className":46478},[75],[19,46480,3484],{"className":46481},[75,76],[19,46483,46485],{"className":46484},[133],[19,46486,46488,46511],{"className":46487},[137,138],[19,46489,46491,46508],{"className":46490},[142],[19,46492,46494],{"className":46493,"style":3609},[146],[19,46495,46496,46499],{"style":150},[19,46497],{"className":46498,"style":155},[154],[19,46500,46502],{"className":46501},[159,160,161,162],[19,46503,46505],{"className":46504},[75,162],[19,46506,3297],{"className":46507},[75,76,162],[19,46509,173],{"className":46510},[172],[19,46512,46514],{"className":46513},[142],[19,46515,46517],{"className":46516,"style":180},[146],[19,46518],{},[19,46520,46522,46525],{"className":46521},[75],[19,46523,49],{"className":46524},[75,76],[19,46526,46528],{"className":46527},[133],[19,46529,46531],{"className":46530},[137],[19,46532,46534],{"className":46533},[142],[19,46535,46537],{"className":46536,"style":3429},[146],[19,46538,46539,46542],{"style":1643},[19,46540],{"className":46541,"style":155},[154],[19,46543,46545],{"className":46544},[159,160,161,162],[19,46546,46548],{"className":46547},[75,162],[19,46549,3297],{"className":46550},[75,76,162]," is continuous on ((-R,R)).",[15,46553,46554,43313,46594,46789],{},[19,46555,46557,46576],{"className":46556,"translate":23},[22],[19,46558,46560],{"className":46559},[27],[29,46561,46562],{"xmlns":31},[33,46563,46564,46574],{},[36,46565,46566,46568,46570,46572],{},[43,46567,46],{"stretchy":45},[39,46569,34238],{},[39,46571,34238],{},[43,46573,52],{"stretchy":45},[54,46575,34491],{"encoding":56},[19,46577,46579],{"className":46578,"ariaHidden":62},[61],[19,46580,46582,46585,46588,46591],{"className":46581},[66],[19,46583],{"className":46584,"style":71},[70],[19,46586,46],{"className":46587},[81],[19,46589,34507],{"className":46590},[75,76],[19,46592,52],{"className":46593},[88],[19,46595,46597,46636],{"className":46596,"translate":23},[22],[19,46598,46600],{"className":46599},[27],[29,46601,46602],{"xmlns":31},[33,46603,46604,46634],{},[36,46605,46606,46620],{},[4285,46607,46608,46610,46618],{},[43,46609,26844],{},[36,46611,46612,46614,46616],{},[39,46613,3297],{},[43,46615,267],{},[108,46617,110],{},[39,46619,26855],{"mathvariant":275},[36,46621,46622,46628],{},[103,46623,46624,46626],{},[39,46625,3484],{},[39,46627,3297],{},[269,46629,46630,46632],{},[39,46631,49],{},[39,46633,3297],{},[54,46635,28591],{"encoding":56},[19,46637,46639],{"className":46638,"ariaHidden":62},[61],[19,46640,46642,46645,46708,46711],{"className":46641},[66],[19,46643],{"className":46644,"style":31929},[70],[19,46646,46648,46651],{"className":46647},[304],[19,46649,26844],{"className":46650,"style":31937},[304,27031,31936],[19,46652,46654],{"className":46653},[133],[19,46655,46657,46700],{"className":46656},[137,138],[19,46658,46660,46697],{"className":46659},[142],[19,46661,46663,46683],{"className":46662,"style":31950},[146],[19,46664,46665,46668],{"style":31953},[19,46666],{"className":46667,"style":155},[154],[19,46669,46671],{"className":46670},[159,160,161,162],[19,46672,46674,46677,46680],{"className":46673},[75,162],[19,46675,3297],{"className":46676},[75,76,162],[19,46678,267],{"className":46679},[335,162],[19,46681,110],{"className":46682},[75,162],[19,46684,46685,46688],{"style":31974},[19,46686],{"className":46687,"style":155},[154],[19,46689,46691],{"className":46690},[159,160,161,162],[19,46692,46694],{"className":46693},[75,162],[19,46695,26855],{"className":46696},[75,162],[19,46698,173],{"className":46699},[172],[19,46701,46703],{"className":46702},[142],[19,46704,46706],{"className":46705,"style":31996},[146],[19,46707],{},[19,46709],{"className":46710,"style":412},[411],[19,46712,46714,46757],{"className":46713},[75],[19,46715,46717,46720],{"className":46716},[75],[19,46718,3484],{"className":46719},[75,76],[19,46721,46723],{"className":46722},[133],[19,46724,46726,46749],{"className":46725},[137,138],[19,46727,46729,46746],{"className":46728},[142],[19,46730,46732],{"className":46731,"style":3609},[146],[19,46733,46734,46737],{"style":150},[19,46735],{"className":46736,"style":155},[154],[19,46738,46740],{"className":46739},[159,160,161,162],[19,46741,46743],{"className":46742},[75,162],[19,46744,3297],{"className":46745},[75,76,162],[19,46747,173],{"className":46748},[172],[19,46750,46752],{"className":46751},[142],[19,46753,46755],{"className":46754,"style":180},[146],[19,46756],{},[19,46758,46760,46763],{"className":46759},[75],[19,46761,49],{"className":46762},[75,76],[19,46764,46766],{"className":46765},[133],[19,46767,46769],{"className":46768},[137],[19,46770,46772],{"className":46771},[142],[19,46773,46775],{"className":46774,"style":3429},[146],[19,46776,46777,46780],{"style":1643},[19,46778],{"className":46779,"style":155},[154],[19,46781,46783],{"className":46782},[159,160,161,162],[19,46784,46786],{"className":46785},[75,162],[19,46787,3297],{"className":46788},[75,76,162]," converges at the left (right) endpoint of its convergence interval, then its sum function is also continuous at this endpoint from the right (left).",[15,46791,46792,46794,46795,46990,46991,47186,47187,47225,47226,46203,47421,43619,47471,43677,47526,43873,47721,43619,47772,47828,47829,47869],{},[7600,46793,16911],{}," Since each term of the power series 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is continuous on ((-R,R)), and ",[19,46992,46994,47033],{"className":46993,"translate":23},[22],[19,46995,46997],{"className":46996},[27],[29,46998,46999],{"xmlns":31},[33,47000,47001,47031],{},[36,47002,47003,47017],{},[4285,47004,47005,47007,47015],{},[43,47006,26844],{},[36,47008,47009,47011,47013],{},[39,47010,3297],{},[43,47012,267],{},[108,47014,110],{},[39,47016,26855],{"mathvariant":275},[36,47018,47019,47025],{},[103,47020,47021,47023],{},[39,47022,3484],{},[39,47024,3297],{},[269,47026,47027,47029],{},[39,47028,49],{},[39,47030,3297],{},[54,47032,28591],{"encoding":56},[19,47034,47036],{"className":47035,"ariaHidden":62},[61],[19,47037,47039,47042,47105,47108],{"className":47038},[66],[19,47040],{"className":47041,"style":31929},[70],[19,47043,47045,47048],{"className":47044},[304],[19,47046,26844],{"className":47047,"style":31937},[304,27031,31936],[19,47049,47051],{"className":47050},[133],[19,47052,47054,47097],{"className":47053},[137,138],[19,47055,47057,47094],{"className":47056},[142],[19,47058,47060,47080],{"className":47059,"style":31950},[146],[19,47061,47062,47065],{"style":31953},[19,47063],{"className":47064,"style":155},[154],[19,47066,47068],{"className":47067},[159,160,161,162],[19,47069,47071,47074,47077],{"className":47070},[75,162],[19,47072,3297],{"className":47073},[75,76,162],[19,47075,267],{"className":47076},[335,162],[19,47078,110],{"className":47079},[75,162],[19,47081,47082,47085],{"style":31974},[19,47083],{"className":47084,"style":155},[154],[19,47086,47088],{"className":47087},[159,160,161,162],[19,47089,47091],{"className":47090},[75,162],[19,47092,26855],{"className":47093},[75,162],[19,47095,173],{"className":47096},[172],[19,47098,47100],{"className":47099},[142],[19,47101,47103],{"className":47102,"style":31996},[146],[19,47104],{},[19,47106],{"className":47107,"style":412},[411],[19,47109,47111,47154],{"className":47110},[75],[19,47112,47114,47117],{"className":47113},[75],[19,47115,3484],{"className":47116},[75,76],[19,47118,47120],{"className":47119},[133],[19,47121,47123,47146],{"className":47122},[137,138],[19,47124,47126,47143],{"className":47125},[142],[19,47127,47129],{"className":47128,"style":3609},[146],[19,47130,47131,47134],{"style":150},[19,47132],{"className":47133,"style":155},[154],[19,47135,47137],{"className":47136},[159,160,161,162],[19,47138,47140],{"className":47139},[75,162],[19,47141,3297],{"className":47142},[75,76,162],[19,47144,173],{"className":47145},[172],[19,47147,47149],{"className":47148},[142],[19,47150,47152],{"className":47151,"style":180},[146],[19,47153],{},[19,47155,47157,47160],{"className":47156},[75],[19,47158,49],{"className":47159},[75,76],[19,47161,47163],{"className":47162},[133],[19,47164,47166],{"className":47165},[137],[19,47167,47169],{"className":47168},[142],[19,47170,47172],{"className":47171,"style":3429},[146],[19,47173,47174,47177],{"style":1643},[19,47175],{"className":47176,"style":155},[154],[19,47178,47180],{"className":47179},[159,160,161,162],[19,47181,47183],{"className":47182},[75,162],[19,47184,3297],{"className":47185},[75,76,162]," uniformly converges on ((-R,R)), we can deduce property ",[19,47188,47190,47207],{"className":47189,"translate":23},[22],[19,47191,47193],{"className":47192},[27],[29,47194,47195],{"xmlns":31},[33,47196,47197,47205],{},[36,47198,47199,47201,47203],{},[43,47200,46],{"stretchy":45},[39,47202,34238],{},[43,47204,52],{"stretchy":45},[54,47206,34243],{"encoding":56},[19,47208,47210],{"className":47209,"ariaHidden":62},[61],[19,47211,47213,47216,47219,47222],{"className":47212},[66],[19,47214],{"className":47215,"style":71},[70],[19,47217,46],{"className":47218},[81],[19,47220,34238],{"className":47221},[75,76],[19,47223,52],{"className":47224},[88]," from the properties of uniform convergence of series of functions. If the power series 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by the same reasoning, thus property ",[19,47830,47832,47851],{"className":47831,"translate":23},[22],[19,47833,47835],{"className":47834},[27],[29,47836,47837],{"xmlns":31},[33,47838,47839,47849],{},[36,47840,47841,47843,47845,47847],{},[43,47842,46],{"stretchy":45},[39,47844,34238],{},[39,47846,34238],{},[43,47848,52],{"stretchy":45},[54,47850,34491],{"encoding":56},[19,47852,47854],{"className":47853,"ariaHidden":62},[61],[19,47855,47857,47860,47863,47866],{"className":47856},[66],[19,47858],{"className":47859,"style":71},[70],[19,47861,46],{"className":47862},[81],[19,47864,34507],{"className":47865},[75,76],[19,47867,52],{"className":47868},[88]," holds.",[15,47871,47872],{},"In the study of series of functions, there are theorems concerning term-by-term integration and differentiation:",[15,47874,47875,47878],{},[7600,47876,47877],{},"(Term-by-term Integration Theorem):"," If the series of functions",[19,47880,47882],{"className":47881,"translate":23},[187],[19,47883,47885,47926],{"className":47884,"translate":23},[22],[19,47886,47888],{"className":47887},[27],[29,47889,47890],{"xmlns":31,"display":196},[33,47891,47892,47923],{},[36,47893,47894,47908],{},[26840,47895,47896,47898,47906],{},[43,47897,26844],{},[36,47899,47900,47902,47904],{},[39,47901,3297],{},[43,47903,267],{},[108,47905,110],{},[39,47907,26855],{"mathvariant":275},[36,47909,47910,47917,47919,47921],{},[103,47911,47912,47915],{},[39,47913,47914],{},"u",[39,47916,3297],{},[43,47918,46],{"stretchy":45},[39,47920,49],{},[43,47922,52],{"stretchy":45},[54,47924,47925],{"encoding":56},"\\sum_{n=0}^{\\infty}{u_{n}(x)}",[19,47927,47929],{"className":47928,"ariaHidden":62},[61],[19,47930,47932,47935,48002,48005],{"className":47931},[66],[19,47933],{"className":47934,"style":26984},[70],[19,47936,47938],{"className":47937},[304,305],[19,47939,47941,47994],{"className":47940},[137,138],[19,47942,47944,47991],{"className":47943},[142],[19,47945,47947,47967,47977],{"className":47946,"style":26997},[146],[19,47948,47949,47952],{"style":27000},[19,47950],{"className":47951,"style":27004},[154],[19,47953,47955],{"className":47954},[159,160,161,162],[19,47956,47958,47961,47964],{"className":47957},[75,162],[19,47959,3297],{"className":47960},[75,76,162],[19,47962,267],{"className":47963},[335,162],[19,47965,110],{"className":47966},[75,162],[19,47968,47969,47972],{"style":27022},[19,47970],{"className":47971,"style":27004},[154],[19,47973,47974],{},[19,47975,26844],{"className":47976},[304,27031,27032],[19,47978,47979,47982],{"style":27035},[19,47980],{"className":47981,"style":27004},[154],[19,47983,47985],{"className":47984},[159,160,161,162],[19,47986,47988],{"className":47987},[75,162],[19,47989,26855],{"className":47990},[75,162],[19,47992,173],{"className":47993},[172],[19,47995,47997],{"className":47996},[142],[19,47998,48000],{"className":47999,"style":27057},[146],[19,48001],{},[19,48003],{"className":48004,"style":412},[411],[19,48006,48008,48051,48054,48057],{"className":48007},[75],[19,48009,48011,48014],{"className":48010},[75],[19,48012,47914],{"className":48013},[75,76],[19,48015,48017],{"className":48016},[133],[19,48018,48020,48043],{"className":48019},[137,138],[19,48021,48023,48040],{"className":48022},[142],[19,48024,48026],{"className":48025,"style":3609},[146],[19,48027,48028,48031],{"style":150},[19,48029],{"className":48030,"style":155},[154],[19,48032,48034],{"className":48033},[159,160,161,162],[19,48035,48037],{"className":48036},[75,162],[19,48038,3297],{"className":48039},[75,76,162],[19,48041,173],{"className":48042},[172],[19,48044,48046],{"className":48045},[142],[19,48047,48049],{"className":48048,"style":180},[146],[19,48050],{},[19,48052,46],{"className":48053},[81],[19,48055,49],{"className":48056},[75,76],[19,48058,52],{"className":48059},[88],[15,48061,43146,48062,48113,48114,48202],{},[19,48063,48065,48086],{"className":48064,"translate":23},[22],[19,48066,48068],{"className":48067},[27],[29,48069,48070],{"xmlns":31},[33,48071,48072,48084],{},[36,48073,48074,48076,48078,48080,48082],{},[43,48075,14194],{"stretchy":45},[39,48077,3484],{},[43,48079,7830],{"separator":62},[39,48081,17223],{},[43,48083,14287],{"stretchy":45},[54,48085,42218],{"encoding":56},[19,48087,48089],{"className":48088,"ariaHidden":62},[61],[19,48090,48092,48095,48098,48101,48104,48107,48110],{"className":48091},[66],[19,48093],{"className":48094,"style":71},[70],[19,48096,14194],{"className":48097},[81],[19,48099,3484],{"className":48100},[75,76],[19,48102,7830],{"className":48103},[8036],[19,48105],{"className":48106,"style":412},[411],[19,48108,17223],{"className":48109},[75,76],[19,48111,14287],{"className":48112},[88],", and each term 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is continuous, 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is a point of convergence 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and",[19,49257,49259],{"className":49258,"translate":23},[187],[19,49260,49262,49304],{"className":49261,"translate":23},[22],[19,49263,49265],{"className":49264},[27],[29,49266,49267],{"xmlns":31,"display":196},[33,49268,49269,49301],{},[36,49270,49271,49285],{},[26840,49272,49273,49275,49283],{},[43,49274,26844],{},[36,49276,49277,49279,49281],{},[39,49278,3297],{},[43,49280,267],{},[108,49282,110],{},[39,49284,26855],{"mathvariant":275},[36,49286,49287,49295,49297,49299],{},[4285,49288,49289,49291,49293],{},[39,49290,47914],{},[39,49292,3297],{},[43,49294,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,49296,46],{"stretchy":45},[39,49298,49],{},[43,49300,52],{"stretchy":45},[54,49302,49303],{"encoding":56},"\\sum_{n=0}^{\\infty}{u'_{n}(x)}",[19,49305,49307],{"className":49306,"ariaHidden":62},[61],[19,49308,49310,49313,49380,49383],{"className":49309},[66],[19,49311],{"className":49312,"style":26984},[70],[19,49314,49316],{"className":49315},[304,305],[19,49317,49319,49372],{"className":49318},[137,138],[19,49320,49322,49369],{"className":49321},[142],[19,49323,49325,49345,49355],{"className":49324,"style":26997},[146],[19,49326,49327,49330],{"style":27000},[19,49328],{"className":49329,"style":27004},[154],[19,49331,49333],{"className":49332},[159,160,161,162],[19,49334,49336,49339,49342],{"className":49335},[75,162],[19,49337,3297],{"className":49338},[75,76,162],[19,49340,267],{"className":49341},[335,162],[19,49343,110],{"className":49344},[75,162],[19,49346,49347,49350],{"style":27022},[19,49348],{"className":49349,"style":27004},[154],[19,49351,49352],{},[19,49353,26844],{"className":49354},[304,27031,27032],[19,49356,49357,49360],{"style":27035},[19,49358],{"className":49359,"style":27004},[154],[19,49361,49363],{"className":49362},[159,160,161,162],[19,49364,49366],{"className":49365},[75,162],[19,49367,26855],{"className":49368},[75,162],[19,49370,173],{"className":49371},[172],[19,49373,49375],{"className":49374},[142],[19,49376,49378],{"className":49377,"style":27057},[146],[19,49379],{},[19,49381],{"className":49382,"style":412},[411],[19,49384,49386,49443,49446,49449],{"className":49385},[75],[19,49387,49389,49392],{"className":49388},[75],[19,49390,47914],{"className":49391},[75,76],[19,49393,49395],{"className":49394},[133],[19,49396,49398,49435],{"className":49397},[137,138],[19,49399,49401,49432],{"className":49400},[142],[19,49402,49404,49418],{"className":49403,"style":645},[146],[19,49405,49406,49409],{"style":12661},[19,49407],{"className":49408,"style":155},[154],[19,49410,49412],{"className":49411},[159,160,161,162],[19,49413,49415],{"className":49414},[75,162],[19,49416,3297],{"className":49417},[75,76,162],[19,49419,49420,49423],{"style":648},[19,49421],{"className":49422,"style":155},[154],[19,49424,49426],{"className":49425},[159,160,161,162],[19,49427,49429],{"className":49428},[75,162],[19,49430,277],{"className":49431},[75,162],[19,49433,173],{"className":49434},[172],[19,49436,49438],{"className":49437},[142],[19,49439,49441],{"className":49440,"style":4621},[146],[19,49442],{},[19,49444,46],{"className":49445},[81],[19,49447,49],{"className":49448},[75,76],[19,49450,52],{"className":49451},[88],[15,49453,43146,49454,9276],{},[19,49455,49457,49478],{"className":49456,"translate":23},[22],[19,49458,49460],{"className":49459},[27],[29,49461,49462],{"xmlns":31},[33,49463,49464,49476],{},[36,49465,49466,49468,49470,49472,49474],{},[43,49467,14194],{"stretchy":45},[39,49469,3484],{},[43,49471,7830],{"separator":62},[39,49473,17223],{},[43,49475,14287],{"stretchy":45},[54,49477,42218],{"encoding":56},[19,49479,49481],{"className":49480,"ariaHidden":62},[61],[19,49482,49484,49487,49490,49493,49496,49499,49502],{"className":49483},[66],[19,49485],{"className":49486,"style":71},[70],[19,49488,14194],{"className":49489},[81],[19,49491,3484],{"className":49492},[75,76],[19,49494,7830],{"className":49495},[8036],[19,49497],{"className":49498,"style":412},[411],[19,49500,17223],{"className":49501},[75,76],[19,49503,14287],{"className":49504},[88],[19,49506,49508],{"className":49507,"translate":23},[187],[19,49509,49511,49609],{"className":49510,"translate":23},[22],[19,49512,49514],{"className":49513},[27],[29,49515,49516],{"xmlns":31,"display":196},[33,49517,49518,49606],{},[36,49519,49520,49534,49536,49560,49562,49564,49574,49576,49590,49604],{},[26840,49521,49522,49524,49532],{},[43,49523,26844],{},[36,49525,49526,49528,49530],{},[39,49527,3297],{},[43,49529,267],{},[108,49531,110],{},[39,49533,26855],{"mathvariant":275},[43,49535,46],{"stretchy":45},[36,49537,49538,49548,49554,49556,49558],{},[226,49539,49540,49542],{},[39,49541,48257],{},[36,49543,49544,49546],{},[39,49545,48257],{},[39,49547,49],{},[103,49549,49550,49552],{},[39,49551,47914],{},[39,49553,3297],{},[43,49555,46],{"stretchy":45},[39,49557,49],{},[43,49559,52],{"stretchy":45},[43,49561,52],{"stretchy":45},[43,49563,267],{},[226,49565,49566,49568],{},[39,49567,48257],{},[36,49569,49570,49572],{},[39,49571,48257]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The requirement that 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the term-by-term differentiation theorem is for the convenience of application. Its equivalent condition is that",[19,50328,50330],{"className":50329,"translate":23},[187],[19,50331,50333,50372],{"className":50332,"translate":23},[22],[19,50334,50336],{"className":50335},[27],[29,50337,50338],{"xmlns":31,"display":196},[33,50339,50340,50370],{},[36,50341,50342,50356],{},[26840,50343,50344,50346,50354],{},[43,50345,26844],{},[36,50347,50348,50350,50352],{},[39,50349,3297],{},[43,50351,267],{},[108,50353,110],{},[39,50355,26855],{"mathvariant":275},[36,50357,50358,50364,50366,50368],{},[103,50359,50360,50362],{},[39,50361,47914],{},[39,50363,3297],{},[43,50365,46],{"stretchy":45},[39,50367,49],{},[43,50369,52],{"stretchy":45},[54,50371,47925],{"encoding":56},[19,50373,50375],{"className":50374,"ariaHidden":62},[61],[19,50376,50378,50381,50448,50451],{"className":50377},[66],[19,50379],{"className":50380,"style":26984},[70],[19,50382,50384],{"className":50383},[304,305],[19,50385,50387,50440],{"className":50386},[137,138],[19,50388,50390,50437],{"className":50389},[142],[19,50391,50393,50413,50423],{"className":50392,"style":26997},[146],[19,50394,50395,50398],{"style":27000},[19,50396],{"className":50397,"style":27004},[154],[19,50399,50401],{"className":50400},[159,160,161,162],[19,50402,50404,50407,50410],{"className":50403},[75,162],[19,50405,3297],{"className":50406},[75,76,162],[19,50408,267],{"className":50409},[335,162],[19,50411,110],{"className":50412},[75,162],[19,50414,50415,50418],{"style":27022},[19,50416],{"className":50417,"style":27004},[154],[19,50419,50420],{},[19,50421,26844],{"className":50422},[304,27031,27032],[19,50424,50425,50428],{"style":27035},[19,50426],{"className":50427,"style":27004},[154],[19,50429,50431],{"className":50430},[159,160,161,162],[19,50432,50434],{"className":50433},[75,162],[19,50435,26855],{"className":50436},[75,162],[19,50438,173],{"className":50439},[172],[19,50441,50443],{"className":50442},[142],[19,50444,50446],{"className":50445,"style":27057},[146],[19,50447],{},[19,50449],{"className":50450,"style":412},[411],[19,50452,50454,50497,50500,50503],{"className":50453},[75],[19,50455,50457,50460],{"className":50456},[75],[19,50458,47914],{"className":50459},[75,76],[19,50461,50463],{"className":50462},[133],[19,50464,50466,50489],{"className":50465},[137,138],[19,50467,50469,50486],{"className":50468},[142],[19,50470,50472],{"className":50471,"style":3609},[146],[19,50473,50474,50477],{"style":150},[19,50475],{"className":50476,"style":155},[154],[19,50478,50480],{"className":50479},[159,160,161,162],[19,50481,50483],{"className":50482},[75,162],[19,50484,3297],{"className":50485},[75,76,162],[19,50487,173],{"className":50488},[172],[19,50490,50492],{"className":50491},[142],[19,50493,50495],{"className":50494,"style":180},[146],[19,50496],{},[19,50498,46],{"className":50499},[81],[19,50501,49],{"className":50502},[75,76],[19,50504,52],{"className":50505},[88],[15,50507,50508,50509,50560,50561,50612],{},"converges on ",[19,50510,50512,50533],{"className":50511,"translate":23},[22],[19,50513,50515],{"className":50514},[27],[29,50516,50517],{"xmlns":31},[33,50518,50519,50531],{},[36,50520,50521,50523,50525,50527,50529],{},[43,50522,14194],{"stretchy":45},[39,50524,3484],{},[43,50526,7830],{"separator":62},[39,50528,17223],{},[43,50530,14287],{"stretchy":45},[54,50532,42218],{"encoding":56},[19,50534,50536],{"className":50535,"ariaHidden":62},[61],[19,50537,50539,50542,50545,50548,50551,50554,50557],{"className":50538},[66],[19,50540],{"className":50541,"style":71},[70],[19,50543,14194],{"className":50544},[81],[19,50546,3484],{"className":50547},[75,76],[19,50549,7830],{"className":50550},[8036],[19,50552],{"className":50553,"style":412},[411],[19,50555,17223],{"className":50556},[75,76],[19,50558,14287],{"className":50559},[88],". Furthermore, the interval ",[19,50562,50564,50585],{"className":50563,"translate":23},[22],[19,50565,50567],{"className":50566},[27],[29,50568,50569],{"xmlns":31},[33,50570,50571,50583],{},[36,50572,50573,50575,50577,50579,50581],{},[43,50574,14194],{"stretchy":45},[39,50576,3484],{},[43,50578,7830],{"separator":62},[39,50580,17223],{},[43,50582,14287],{"stretchy":45},[54,50584,42218],{"encoding":56},[19,50586,50588],{"className":50587,"ariaHidden":62},[61],[19,50589,50591,50594,50597,50600,50603,50606,50609],{"className":50590},[66],[19,50592],{"className":50593,"style":71},[70],[19,50595,14194],{"className":50596},[81],[19,50598,3484],{"className":50599},[75,76],[19,50601,7830],{"className":50602},[8036],[19,50604],{"className":50605,"style":412},[411],[19,50607,17223],{"className":50608},[75,76],[19,50610,14287],{"className":50611},[88]," in the conditions can be replaced with a general open interval, and the conclusion still holds.",[15,50614,50615],{},"For the power series",[19,50617,50619],{"className":50618,"translate":23},[187],[19,50620,50622,50661],{"className":50621,"translate":23},[22],[19,50623,50625],{"className":50624},[27],[29,50626,50627],{"xmlns":31,"display":196},[33,50628,50629,50659],{},[36,50630,50631,50645],{},[26840,50632,50633,50635,50643],{},[43,50634,26844],{},[36,50636,50637,50639,50641],{},[39,50638,3297],{},[43,50640,267],{},[108,50642,110],{},[39,50644,26855],{"mathvariant":275},[36,50646,50647,50653],{},[103,50648,50649,50651],{},[39,50650,3484],{},[39,50652,3297],{},[269,50654,50655,50657],{},[39,50656,49],{},[39,50658,3297],{},[54,50660,28591],{"encoding":56},[19,50662,50664],{"className":50663,"ariaHidden":62},[61],[19,50665,50667,50670,50737,50740],{"className":50666},[66],[19,50668],{"className":50669,"style":26984},[70],[19,50671,50673],{"className":50672},[304,305],[19,50674,50676,50729],{"className":50675},[137,138],[19,50677,50679,50726],{"className":50678},[142],[19,50680,50682,50702,50712],{"className":50681,"style":26997},[146],[19,50683,50684,50687],{"style":27000},[19,50685],{"className":50686,"style":27004},[154],[19,50688,50690],{"className":50689},[159,160,161,162],[19,50691,50693,50696,50699],{"className":50692},[75,162],[19,50694,3297],{"className":50695},[75,76,162],[19,50697,267],{"className":50698},[335,162],[19,50700,110],{"className":50701},[75,162],[19,50703,50704,50707],{"style":27022},[19,50705],{"className":50706,"style":27004},[154],[19,50708,50709],{},[19,50710,26844],{"className":50711},[304,27031,27032],[19,50713,50714,50717],{"style":27035},[19,50715],{"className":50716,"style":27004},[154],[19,50718,50720],{"className":50719},[159,160,161,162],[19,50721,50723],{"className":50722},[75,162],[19,50724,26855],{"className":50725},[75,162],[19,50727,173],{"className":50728},[172],[19,50730,50732],{"className":50731},[142],[19,50733,50735],{"className":50734,"style":27057},[146],[19,50736],{},[19,50738],{"className":50739,"style":412},[411],[19,50741,50743,50786],{"className":50742},[75],[19,50744,50746,50749],{"className":50745},[75],[19,50747,3484],{"className":50748},[75,76],[19,50750,50752],{"className":50751},[133],[19,50753,50755,50778],{"className":50754},[137,138],[19,50756,50758,50775],{"className":50757},[142],[19,50759,50761],{"className":50760,"style":3609},[146],[19,50762,50763,50766],{"style":150},[19,50764],{"className":50765,"style":155},[154],[19,50767,50769],{"className":50768},[159,160,161,162],[19,50770,50772],{"className":50771},[75,162],[19,50773,3297],{"className":50774},[75,76,162],[19,50776,173],{"className":50777},[172],[19,50779,50781],{"className":50780},[142],[19,50782,50784],{"className":50783,"style":180},[146],[19,50785],{},[19,50787,50789,50792],{"className":50788},[75],[19,50790,49],{"className":50791},[75,76],[19,50793,50795],{"className":50794},[133],[19,50796,50798],{"className":50797},[137],[19,50799,50801],{"className":50800},[142],[19,50802,50804],{"className":50803,"style":4148},[146],[19,50805,50806,50809],{"style":648},[19,50807],{"className":50808,"style":155},[154],[19,50810,50812],{"className":50811},[159,160,161,162],[19,50813,50815],{"className":50814},[75,162],[19,50816,3297],{"className":50817},[75,76,162],[15,50819,50820],{},"its power series obtained by differentiating each term is",[19,50822,50824],{"className":50823,"translate":23},[187],[19,50825,50827,50875],{"className":50826,"translate":23},[22],[19,50828,50830],{"className":50829},[27],[29,50831,50832],{"xmlns":31,"display":196},[33,50833,50834,50872],{},[36,50835,50836,50850],{},[26840,50837,50838,50840,50848],{},[43,50839,26844],{},[36,50841,50842,50844,50846],{},[39,50843,3297],{},[43,50845,267],{},[108,50847,3495],{},[39,50849,26855],{"mathvariant":275},[36,50851,50852,50854,50860],{},[39,50853,3297],{},[103,50855,50856,50858],{},[39,50857,3484],{},[39,50859,3297],{},[269,50861,50862,50864],{},[39,50863,49],{},[36,50865,50866,50868,50870],{},[39,50867,3297],{},[43,50869,240],{},[108,50871,3495],{},[54,50873,50874],{"encoding":56},"\\sum_{n=1}^{\\infty}{na_{n}x^{n-1}}",[19,50876,50878],{"className":50877,"ariaHidden":62},[61],[19,50879,50881,50884,50951,50954],{"className":50880},[66],[19,50882],{"className":50883,"style":26984},[70],[19,50885,50887],{"className":50886},[304,305],[19,50888,50890,50943],{"className":50889},[137,138],[19,50891,50893,50940],{"className":50892},[142],[19,50894,50896,50916,50926],{"className":50895,"style":26997},[146],[19,50897,50898,50901],{"style":27000},[19,50899],{"className":50900,"style":27004},[154],[19,50902,50904],{"className":50903},[159,160,161,162],[19,50905,50907,50910,50913],{"className":50906},[75,162],[19,50908,3297],{"className":50909},[75,76,162],[19,50911,267],{"className":50912},[335,162],[19,50914,3495],{"className":50915},[75,162],[19,50917,50918,50921],{"style":27022},[19,50919],{"className":50920,"style":27004},[154],[19,50922,50923],{},[19,50924,26844],{"className":50925},[304,27031,27032],[19,50927,50928,50931],{"style":27035},[19,50929],{"className":50930,"style":27004},[154],[19,50932,50934],{"className":50933},[159,160,161,162],[19,50935,50937],{"className":50936},[75,162],[19,50938,26855],{"className":50939},[75,162],[19,50941,173],{"className":50942},[172],[19,50944,50946],{"className":50945},[142],[19,50947,50949],{"className":50948,"style":27057},[146],[19,50950],{},[19,50952],{"className":50953,"style":412},[411],[19,50955,50957,50960,51003],{"className":50956},[75],[19,50958,3297],{"className":50959},[75,76],[19,50961,50963,50966],{"className":50962},[75],[19,50964,3484],{"className":50965},[75,76],[19,50967,50969],{"className":50968},[133],[19,50970,50972,50995],{"className":50971},[137,138],[19,50973,50975,50992],{"className":50974},[142],[19,50976,50978],{"className":50977,"style":3609},[146],[19,50979,50980,50983],{"style":150},[19,50981],{"className":50982,"style":155},[154],[19,50984,50986],{"className":50985},[159,160,161,162],[19,50987,50989],{"className":50988},[75,162],[19,50990,3297],{"className":50991},[75,76,162],[19,50993,173],{"className":50994},[172],[19,50996,50998],{"className":50997},[142],[19,50999,51001],{"className":51000,"style":180},[146],[19,51002],{},[19,51004,51006,51009],{"className":51005},[75],[19,51007,49],{"className":51008},[75,76],[19,51010,51012],{"className":51011},[133],[19,51013,51015],{"className":51014},[137],[19,51016,51018],{"className":51017},[142],[19,51019,51021],{"className":51020,"style":3970},[146],[19,51022,51023,51026],{"style":648},[19,51024],{"className":51025,"style":155},[154],[19,51027,51029],{"className":51028},[159,160,161,162],[19,51030,51032,51035,51038],{"className":51031},[75,162],[19,51033,3297],{"className":51034},[75,76,162],[19,51036,240],{"className":51037},[455,162],[19,51039,3495],{"className":51040},[75,162],[15,51042,51043],{},"and the power series obtained by integrating each term is",[19,51045,51047],{"className":51046,"translate":23},[187],[19,51048,51050,51106],{"className":51049,"translate":23},[22],[19,51051,51053],{"className":51052},[27],[29,51054,51055],{"xmlns":31,"display":196},[33,51056,51057,51103],{},[36,51058,51059,51073],{},[26840,51060,51061,51063,51071],{},[43,51062,26844],{},[36,51064,51065,51067,51069],{},[39,51066,3297],{},[43,51068,267],{},[108,51070,110],{},[39,51072,26855],{"mathvariant":275},[226,51074,51075,51095],{},[36,51076,51077,51083],{},[103,51078,51079,51081],{},[39,51080,3484],{},[39,51082,3297],{},[269,51084,51085,51087],{},[39,51086,49],{},[36,51088,51089,51091,51093],{},[39,51090,3297],{},[43,51092,2351],{},[108,51094,3495],{},[36,51096,51097,51099,51101],{},[39,51098,3297],{},[43,51100,2351],{},[108,51102,3495],{},[54,51104,51105],{"encoding":56},"\\sum_{n=0}^{\\infty}{\\frac{a_{n}x^{n+1}}{n+1}}",[19,51107,51109],{"className":51108,"ariaHidden":62},[61],[19,51110,51112,51115,51182,51185],{"className":51111},[66],[19,51113],{"className":51114,"style":26984},[70],[19,51116,51118],{"className":51117},[304,305],[19,51119,51121,51174],{"className":51120},[137,138],[19,51122,51124,51171],{"className":51123},[142],[19,51125,51127,51147,51157],{"className":51126,"style":26997},[146],[19,51128,51129,51132],{"style":27000},[19,51130],{"className":51131,"style":27004},[154],[19,51133,51135],{"className":51134},[159,160,161,162],[19,51136,51138,51141,51144],{"className":51137},[75,162],[19,51139,3297],{"className":51140},[75,76,162],[19,51142,267],{"className":51143},[335,162],[19,51145,110],{"className":51146},[75,162],[19,51148,51149,51152],{"style":27022},[19,51150],{"className":51151,"style":27004},[154],[19,51153,51154],{},[19,51155,26844],{"className":51156},[304,27031,27032],[19,51158,51159,51162],{"style":27035},[19,51160],{"className":51161,"style":27004},[154],[19,51163,51165],{"className":51164},[159,160,161,162],[19,51166,51168],{"className":51167},[75,162],[19,51169,26855],{"className":51170},[75,162],[19,51172,173],{"className":51173},[172],[19,51175,51177],{"className":51176},[142],[19,51178,51180],{"className":51179,"style":27057},[146],[19,51181],{},[19,51183],{"className":51184,"style":412},[411],[19,51186,51188],{"className":51187},[75],[19,51189,51191,51194,51337],{"className":51190},[75],[19,51192],{"className":51193},[81,422],[19,51195,51197],{"className":51196},[226],[19,51198,51200,51329],{"className":51199},[137,138],[19,51201,51203,51326],{"className":51202},[142],[19,51204,51206,51229,51237],{"className":51205,"style":39338},[146],[19,51207,51208,51211],{"style":438},[19,51209],{"className":51210,"style":322},[154],[19,51212,51214,51217,51220,51223,51226],{"className":51213},[75],[19,51215,3297],{"className":51216},[75,76],[19,51218],{"className":51219,"style":451},[411],[19,51221,2351],{"className":51222},[455],[19,51224],{"className":51225,"style":451},[411],[19,51227,3495],{"className":51228},[75],[19,51230,51231,51234],{"style":504},[19,51232],{"className":51233,"style":322},[154],[19,51235],{"className":51236,"style":512},[511],[19,51238,51239,51242],{"style":515},[19,51240],{"className":51241,"style":322},[154],[19,51243,51245,51288],{"className":51244},[75],[19,51246,51248,51251],{"className":51247},[75],[19,51249,3484],{"className":51250},[75,76],[19,51252,51254],{"className":51253},[133],[19,51255,51257,51280],{"className":51256},[137,138],[19,51258,51260,51277],{"className":51259},[142],[19,51261,51263],{"className":51262,"style":3609},[146],[19,51264,51265,51268],{"style":150},[19,51266],{"className":51267,"style":155},[154],[19,51269,51271],{"className":51270},[159,160,161,162],[19,51272,51274],{"className":51273},[75,162],[19,51275,3297],{"className":51276},[75,76,162],[19,51278,173],{"className":51279},[172],[19,51281,51283],{"className":51282},[142],[19,51284,51286],{"className":51285,"style":180},[146],[19,51287],{},[19,51289,51291,51294],{"className":51290},[75],[19,51292,49],{"className":51293},[75,76],[19,51295,51297],{"className":51296},[133],[19,51298,51300],{"className":51299},[137],[19,51301,51303],{"className":51302},[142],[19,51304,51306],{"className":51305,"style":19038},[146],[19,51307,51308,51311],{"style":1643},[19,51309],{"className":51310,"style":155},[154],[19,51312,51314],{"className":51313},[159,160,161,162],[19,51315,51317,51320,51323],{"className":51316},[75,162],[19,51318,3297],{"className":51319},[75,76,162],[19,51321,2351],{"className":51322},[455,162],[19,51324,3495],{"className":51325},[75,162],[19,51327,173],{"className":51328},[172],[19,51330,51332],{"className":51331},[142],[19,51333,51335],{"className":51334,"style":39455},[146],[19,51336],{},[19,51338],{"className":51339},[88,422],[15,51341,51342,51343,13307,51382,21529,51421,51460],{},"For convenience, let's denote the three power series as ",[19,51344,51346,51364],{"className":51345,"translate":23},[22],[19,51347,51349],{"className":51348},[27],[29,51350,51351],{"xmlns":31},[33,51352,51353,51361],{},[36,51354,51355,51357,51359],{},[43,51356,46],{"stretchy":45},[108,51358,3495],{},[43,51360,52],{"stretchy":45},[54,51362,51363],{"encoding":56},"(1)",[19,51365,51367],{"className":51366,"ariaHidden":62},[61],[19,51368,51370,51373,51376,51379],{"className":51369},[66],[19,51371],{"className":51372,"style":71},[70],[19,51374,46],{"className":51375},[81],[19,51377,3495],{"className":51378},[75],[19,51380,52],{"className":51381},[88],[19,51383,51385,51403],{"className":51384,"translate":23},[22],[19,51386,51388],{"className":51387},[27],[29,51389,51390],{"xmlns":31},[33,51391,51392,51400],{},[36,51393,51394,51396,51398],{},[43,51395,46],{"stretchy":45},[108,51397,3518],{},[43,51399,52],{"stretchy":45},[54,51401,51402],{"encoding":56},"(2)",[19,51404,51406],{"className":51405,"ariaHidden":62},[61],[19,51407,51409,51412,51415,51418],{"className":51408},[66],[19,51410],{"className":51411,"style":71},[70],[19,51413,46],{"className":51414},[81],[19,51416,3518],{"className":51417},[75],[19,51419,52],{"className":51420},[88],[19,51422,51424,51442],{"className":51423,"translate":23},[22],[19,51425,51427],{"className":51426},[27],[29,51428,51429],{"xmlns":31},[33,51430,51431,51439],{},[36,51432,51433,51435,51437],{},[43,51434,46],{"stretchy":45},[108,51436,39228],{},[43,51438,52],{"stretchy":45},[54,51440,51441],{"encoding":56},"(3)",[19,51443,51445],{"className":51444,"ariaHidden":62},[61],[19,51446,51448,51451,51454,51457],{"className":51447},[66],[19,51449],{"className":51450,"style":71},[70],[19,51452,46],{"className":51453},[81],[19,51455,39228],{"className":51456},[75],[19,51458,52],{"className":51459},[88],", respectively.",[15,51462,51463],{},"To discuss term-by-term differentiation and integration of power series, we first introduce the following lemma:",[15,51465,51466,51469,51470,13643,51508,13307,51546,51584],{},[7600,51467,51468],{},"Lemma:"," Power series ",[19,51471,51473,51490],{"className":51472,"translate":23},[22],[19,51474,51476],{"className":51475},[27],[29,51477,51478],{"xmlns":31},[33,51479,51480,51488],{},[36,51481,51482,51484,51486],{},[43,51483,46],{"stretchy":45},[108,51485,3495],{},[43,51487,52],{"stretchy":45},[54,51489,51363],{"encoding":56},[19,51491,51493],{"className":51492,"ariaHidden":62},[61],[19,51494,51496,51499,51502,51505],{"className":51495},[66],[19,51497],{"className":51498,"style":71},[70],[19,51500,46],{"className":51501},[81],[19,51503,3495],{"className":51504},[75],[19,51506,52],{"className":51507},[88],[19,51509,51511,51528],{"className":51510,"translate":23},[22],[19,51512,51514],{"className":51513},[27],[29,51515,51516],{"xmlns":31},[33,51517,51518,51526],{},[36,51519,51520,51522,51524],{},[43,51521,46],{"stretchy":45},[108,51523,3518],{},[43,51525,52],{"stretchy":45},[54,51527,51402],{"encoding":56},[19,51529,51531],{"className":51530,"ariaHidden":62},[61],[19,51532,51534,51537,51540,51543],{"className":51533},[66],[19,51535],{"className":51536,"style":71},[70],[19,51538,46],{"className":51539},[81],[19,51541,3518],{"className":51542},[75],[19,51544,52],{"className":51545},[88],[19,51547,51549,51566],{"className":51548,"translate":23},[22],[19,51550,51552],{"className":51551},[27],[29,51553,51554],{"xmlns":31},[33,51555,51556,51564],{},[36,51557,51558,51560,51562],{},[43,51559,46],{"stretchy":45},[108,51561,39228],{},[43,51563,52],{"stretchy":45},[54,51565,51441],{"encoding":56},[19,51567,51569],{"className":51568,"ariaHidden":62},[61],[19,51570,51572,51575,51578,51581],{"className":51571},[66],[19,51573],{"className":51574,"style":71},[70],[19,51576,46],{"className":51577},[81],[19,51579,39228],{"className":51580},[75],[19,51582,52],{"className":51583},[88]," have the same interval of convergence.",[15,51586,51587,51589,51590,13643,51628,51666,51667,51705,51706,3447],{},[7600,51588,10737],{}," We only need to prove that ",[19,51591,51593,51610],{"className":51592,"translate":23},[22],[19,51594,51596],{"className":51595},[27],[29,51597,51598],{"xmlns":31},[33,51599,51600,51608],{},[36,51601,51602,51604,51606],{},[43,51603,46],{"stretchy":45},[108,51605,3495],{},[43,51607,52],{"stretchy":45},[54,51609,51363],{"encoding":56},[19,51611,51613],{"className":51612,"ariaHidden":62},[61],[19,51614,51616,51619,51622,51625],{"className":51615},[66],[19,51617],{"className":51618,"style":71},[70],[19,51620,46],{"className":51621},[81],[19,51623,3495],{"className":51624},[75],[19,51626,52],{"className":51627},[88],[19,51629,51631,51648],{"className":51630,"translate":23},[22],[19,51632,51634],{"className":51633},[27],[29,51635,51636],{"xmlns":31},[33,51637,51638,51646],{},[36,51639,51640,51642,51644],{},[43,51641,46],{"stretchy":45},[108,51643,3518],{},[43,51645,52],{"stretchy":45},[54,51647,51402],{"encoding":56},[19,51649,51651],{"className":51650,"ariaHidden":62},[61],[19,51652,51654,51657,51660,51663],{"className":51653},[66],[19,51655],{"className":51656,"style":71},[70],[19,51658,46],{"className":51659},[81],[19,51661,3518],{"className":51662},[75],[19,51664,52],{"className":51665},[88]," have the same interval of convergence because differentiating each term of ",[19,51668,51670,51687],{"className":51669,"translate":23},[22],[19,51671,51673],{"className":51672},[27],[29,51674,51675],{"xmlns":31},[33,51676,51677,51685],{},[36,51678,51679,51681,51683],{},[43,51680,46],{"stretchy":45},[108,51682,39228],{},[43,51684,52],{"stretchy":45},[54,51686,51441],{"encoding":56},[19,51688,51690],{"className":51689,"ariaHidden":62},[61],[19,51691,51693,51696,51699,51702],{"className":51692},[66],[19,51694],{"className":51695,"style":71},[70],[19,51697,46],{"className":51698},[81],[19,51700,39228],{"className":51701},[75],[19,51703,52],{"className":51704},[88]," results in ",[19,51707,51709,51726],{"className":51708,"translate":23},[22],[19,51710,51712],{"className":51711},[27],[29,51713,51714],{"xmlns":31},[33,51715,51716,51724],{},[36,51717,51718,51720,51722],{},[43,51719,46],{"stretchy":45},[108,51721,3518],{},[43,51723,52],{"stretchy":45},[54,51725,51402],{"encoding":56},[19,51727,51729],{"className":51728,"ariaHidden":62},[61],[19,51730,51732,51735,51738,51741],{"className":51731},[66],[19,51733],{"className":51734,"style":71},[70],[19,51736,46],{"className":51737},[81],[19,51739,3518],{"className":51740},[75],[19,51742,52],{"className":51743},[88],[15,51745,51746,51747,51785,51786,51837,51838,46203,51876,51961,51962,52102],{},"Suppose ",[19,51748,51750,51767],{"className":51749,"translate":23},[22],[19,51751,51753],{"className":51752},[27],[29,51754,51755],{"xmlns":31},[33,51756,51757,51765],{},[36,51758,51759,51761,51763],{},[43,51760,46],{"stretchy":45},[108,51762,3495],{},[43,51764,52],{"stretchy":45},[54,51766,51363],{"encoding":56},[19,51768,51770],{"className":51769,"ariaHidden":62},[61],[19,51771,51773,51776,51779,51782],{"className":51772},[66],[19,51774],{"className":51775,"style":71},[70],[19,51777,46],{"className":51778},[81],[19,51780,3495],{"className":51781},[75],[19,51783,52],{"className":51784},[88]," converges only at ",[19,51787,51789,51807],{"className":51788,"translate":23},[22],[19,51790,51792],{"className":51791},[27],[29,51793,51794],{"xmlns":31},[33,51795,51796,51804],{},[36,51797,51798,51800,51802],{},[39,51799,49],{},[43,51801,267],{},[108,51803,110],{},[54,51805,51806],{"encoding":56},"x=0",[19,51808,51810,51828],{"className":51809,"ariaHidden":62},[61],[19,51811,51813,51816,51819,51822,51825],{"className":51812},[66],[19,51814],{"className":51815,"style":3309},[70],[19,51817,49],{"className":51818},[75,76],[19,51820],{"className":51821,"style":613},[411],[19,51823,267],{"className":51824},[335],[19,51826],{"className":51827,"style":613},[411],[19,51829,51831,51834],{"className":51830},[66],[19,51832],{"className":51833,"style":1214},[70],[19,51835,110],{"className":51836},[75],". Assuming ",[19,51839,51841,51858],{"className":51840,"translate":23},[22],[19,51842,51844],{"className":51843},[27],[29,51845,51846],{"xmlns":31},[33,51847,51848,51856],{},[36,51849,51850,51852,51854],{},[43,51851,46],{"stretchy":45},[108,51853,3518],{},[43,51855,52],{"stretchy":45},[54,51857,51402],{"encoding":56},[19,51859,51861],{"className":51860,"ariaHidden":62},[61],[19,51862,51864,51867,51870,51873],{"className":51863},[66],[19,51865],{"className":51866,"style":71},[70],[19,51868,46],{"className":51869},[81],[19,51871,3518],{"className":51872},[75],[19,51874,52],{"className":51875},[88],[19,51877,51879,51901],{"className":51878,"translate":23},[22],[19,51880,51882],{"className":51881},[27],[29,51883,51884],{"xmlns":31},[33,51885,51886,51898],{},[36,51887,51888,51894,51896],{},[269,51889,51890,51892],{},[39,51891,49],{},[43,51893,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,51895,29448],{},[108,51897,110],{},[54,51899,51900],{"encoding":56},"x'>0",[19,51902,51904,51952],{"className":51903,"ariaHidden":62},[61],[19,51905,51907,51911,51943,51946,51949],{"className":51906},[66],[19,51908],{"className":51909,"style":51910},[70],"height:0.791em;vertical-align:-0.0391em;",[19,51912,51914,51917],{"className":51913},[75],[19,51915,49],{"className":51916},[75,76],[19,51918,51920],{"className":51919},[133],[19,51921,51923],{"className":51922},[137],[19,51924,51926],{"className":51925},[142],[19,51927,51929],{"className":51928,"style":1640},[146],[19,51930,51931,51934],{"style":1643},[19,51932],{"className":51933,"style":155},[154],[19,51935,51937],{"className":51936},[159,160,161,162],[19,51938,51940],{"className":51939},[75,162],[19,51941,277],{"className":51942},[75,162],[19,51944],{"className":51945,"style":613},[411],[19,51947,29448],{"className":51948},[335],[19,51950],{"className":51951,"style":613},[411],[19,51953,51955,51958],{"className":51954},[66],[19,51956],{"className":51957,"style":1214},[70],[19,51959,110],{"className":51960},[75],", then for 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by the Abel's theorem,",[19,52104,52106],{"className":52105,"translate":23},[187],[19,52107,52109,52165],{"className":52108,"translate":23},[22],[19,52110,52112],{"className":52111},[27],[29,52113,52114],{"xmlns":31,"display":196},[33,52115,52116,52162],{},[36,52117,52118,52132],{},[26840,52119,52120,52122,52130],{},[43,52121,26844],{},[36,52123,52124,52126,52128],{},[39,52125,3297],{},[43,52127,267],{},[108,52129,3495],{},[39,52131,26855],{"mathvariant":275},[36,52133,52134,52136,52138,52144,52160],{},[43,52135,29161],{"fence":62},[39,52137,3297],{},[103,52139,52140,52142],{},[39,52141,3484],{},[39,52143,3297],{},[269,52145,52146,52152],{},[28980,52147,52148,52150],{"accent":62},[39,52149,49],{},[43,52151,28986],{},[36,52153,52154,52156,52158],{},[39,52155,3297],{},[43,52157,240],{},[108,52159,3495],{},[43,52161,29161],{"fence":62},[54,52163,52164],{"encoding":56},"\\sum_{n=1}^{\\infty}{\\left| na_{n}\\bar{x}^{n-1} \\right|}",[19,52166,52168],{"className":52167,"ariaHidden":62},[61],[19,52169,52171,52174,52241,52244],{"className":52170},[66],[19,52172],{"className":52173,"style":26984},[70],[19,52175,52177],{"className":52176},[304,305],[19,52178,52180,52233],{"className":52179},[137,138],[19,52181,52183,52230],{"className":52182},[142],[19,52184,52186,52206,52216],{"className":52185,"style":26997},[146],[19,52187,52188,52191],{"style":27000},[19,52189],{"className":52190,"style":27004},[154],[19,52192,52194],{"className":52193},[159,160,161,162],[19,52195,52197,52200,52203],{"className":52196},[75,162],[19,52198,3297],{"className":52199},[75,76,162],[19,52201,267],{"className":52202},[335,162],[19,52204,3495],{"className":52205},[75,162],[19,52207,52208,52211],{"style":27022},[19,52209],{"className":52210,"style":27004},[154],[19,52212,52213],{},[19,52214,26844],{"className":52215},[304,27031,27032],[19,52217,52218,52221],{"style":27035},[19,52219],{"className":52220,"style":27004},[154],[19,52222,52224],{"className":52223},[159,160,161,162],[19,52225,52227],{"className":52226},[75,162],[19,52228,26855],{"className":52229},[75,162],[19,52231,173],{"className":52232},[172],[19,52234,52236],{"className":52235},[142],[19,52237,52239],{"className":52238,"style":27057},[146],[19,52240],{},[19,52242],{"className":52243,"style":412},[411],[19,52245,52247],{"className":52246},[75],[19,52248,52250,52287,52290,52333,52399],{"className":52249},[16026],[19,52251,52253],{"className":52252},[81],[19,52254,52256],{"className":52255},[16035,30439],[19,52257,52259,52279],{"className":52258},[137,138],[19,52260,52262,52276],{"className":52261},[142],[19,52263,52265],{"className":52264,"style":30449},[146],[19,52266,52267,52270],{"style":30452},[19,52268],{"className":52269,"style":30456},[154],[19,52271,52272],{"style":30459},[30461,52273,52274],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,52275],{"d":30470},[19,52277,173],{"className":52278},[172],[19,52280,52282],{"className":52281},[142],[19,52283,52285],{"className":52284,"style":30480},[146],[19,52286],{},[19,52288,3297],{"className":52289},[75,76],[19,52291,52293,52296],{"className":52292},[75],[19,52294,3484],{"className":52295},[75,76],[19,52297,52299],{"className":52298},[133],[19,52300,52302,52325],{"className":52301},[137,138],[19,52303,52305,52322],{"className":52304},[142],[19,52306,52308],{"className":52307,"style":3609},[146],[19,52309,52310,52313],{"style":150},[19,52311],{"className":52312,"style":155},[154],[19,52314,52316],{"className":52315},[159,160,161,162],[19,52317,52319],{"className":52318},[75,162],[19,52320,3297],{"className":52321},[75,76,162],[19,52323,173],{"className":52324},[172],[19,52326,52328],{"className":52327},[142],[19,52329,52331],{"className":52330,"style":180},[146],[19,52332],{},[19,52334,52336,52367],{"className":52335},[75],[19,52337,52339],{"className":52338},[75,29025],[19,52340,52342],{"className":52341},[137],[19,52343,52345],{"className":52344},[142],[19,52346,52348,52356],{"className":52347,"style":29035},[146],[19,52349,52350,52353],{"style":387},[19,52351],{"className":52352,"style":322},[154],[19,52354,49],{"className":52355},[75,76],[19,52357,52358,52361],{"style":387},[19,52359],{"className":52360,"style":322},[154],[19,52362,52364],{"className":52363,"style":29053},[29052],[19,52365,28986],{"className":52366},[75],[19,52368,52370],{"className":52369},[133],[19,52371,52373],{"className":52372},[137],[19,52374,52376],{"className":52375},[142],[19,52377,52379],{"className":52378,"style":3970},[146],[19,52380,52381,52384],{"style":648},[19,52382],{"className":52383,"style":155},[154],[19,52385,52387],{"className":52386},[159,160,161,162],[19,52388,52390,52393,52396],{"className":52389},[75,162],[19,52391,3297],{"className":52392},[75,76,162],[19,52394,240],{"className":52395},[455,162],[19,52397,3495],{"className":52398},[75,162],[19,52400,52402],{"className":52401},[88],[19,52403,52405],{"className":52404},[16035,30439],[19,52406,52408,52428],{"className":52407},[137,138],[19,52409,52411,52425],{"className":52410},[142],[19,52412,52414],{"className":52413,"style":30449},[146],[19,52415,52416,52419],{"style":30452},[19,52417],{"className":52418,"style":30456},[154],[19,52420,52421],{"style":30459},[30461,52422,52423],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,52424],{"d":30470},[19,52426,173],{"className":52427},[172],[19,52429,52431],{"className":52430},[142],[19,52432,52434],{"className":52433,"style":30480},[146],[19,52435],{},[15,52437,52438],{},"converges. Thus,",[19,52440,52442],{"className":52441,"translate":23},[187],[19,52443,52445,52569],{"className":52444,"translate":23},[22],[19,52446,52448],{"className":52447},[27],[29,52449,52450],{"xmlns":31,"display":196},[33,52451,52452,52566],{},[36,52453,52454,52476,52478,52508,52524,52526,52556],{},[36,52455,52456,52458,52464,52474],{},[43,52457,29161],{"fence":62},[103,52459,52460,52462],{},[39,52461,3484],{},[39,52463,3297],{},[269,52465,52466,52472],{},[28980,52467,52468,52470],{"accent":62},[39,52469,49],{},[43,52471,28986],{},[39,52473,3297],{},[43,52475,29161],{"fence":62},[43,52477,267],{},[36,52479,52480,52482,52484,52490,52506],{},[43,52481,29161],{"fence":62},[39,52483,3297],{},[103,52485,52486,52488],{},[39,52487,3484],{},[39,52489,3297],{},[269,52491,52492,52498],{},[28980,52493,52494,52496],{"accent":62},[39,52495,49],{},[43,52497,28986],{},[36,52499,52500,52502,52504],{},[39,52501,3297],{},[43,52503,240],{},[108,52505,3495],{},[43,52507,29161],{"fence":62},[36,52509,52510,52512,52522],{},[43,52511,29161],{"fence":62},[226,52513,52514,52520],{},[28980,52515,52516,52518],{"accent":62},[39,52517,49],{},[43,52519,28986],{},[39,52521,3297],{},[43,52523,29161],{"fence":62},[43,52525,267],{},[36,52527,52528,52530,52532,52538,52554],{},[43,52529,29161],{"fence":62},[39,52531,3297],{},[103,52533,52534,52536],{},[39,52535,3484],{},[39,52537,3297],{},[269,52539,52540,52546],{},[28980,52541,52542,52544],{"accent":62},[39,52543,49],{},[43,52545,28986],{},[36,52547,52548,52550,52552],{},[39,52549,3297],{},[43,52551,240],{},[108,52553,3495],{},[43,52555,29161],{"fence":62},[226,52557,52558,52564],{},[28980,52559,52560,52562],{"accent":62},[39,52561,49],{},[43,52563,28986],{},[39,52565,3297],{},[54,52567,52568],{"encoding":56},"\\left| 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\\right\\}",[19,53734,53736],{"className":53735,"ariaHidden":62},[61],[19,53737,53739,53742],{"className":53738},[66],[19,53740],{"className":53741,"style":52705},[70],[19,53743,53745,53751,53841],{"className":53744},[16026],[19,53746,53748],{"className":53747,"style":16031},[81,16030],[19,53749,26641],{"className":53750},[16035,44882],[19,53752,53754,53757,53838],{"className":53753},[75],[19,53755],{"className":53756},[81,422],[19,53758,53760],{"className":53759},[226],[19,53761,53763,53830],{"className":53762},[137,138],[19,53764,53766,53827],{"className":53765},[142],[19,53767,53769,53780,53788],{"className":53768,"style":52956},[146],[19,53770,53771,53774],{"style":438},[19,53772],{"className":53773,"style":322},[154],[19,53775,53777],{"className":53776},[75],[19,53778,3297],{"className":53779},[75,76],[19,53781,53782,53785],{"style":504},[19,53783],{"className":53784,"style":322},[154],[19,53786],{"className":53787,"style":512},[511],[19,53789,53790,53793],{"style":515},[19,53791],{"className":53792,"style":322},[154],[19,53794,53796],{"className":53795},[75],[19,53797,53799],{"className":53798},[75,29025],[19,53800,53802],{"className":53801},[137],[19,53803,53805],{"className":53804},[142],[19,53806,53808,53816],{"className":53807,"style":29035},[146],[19,53809,53810,53813],{"style":387},[19,53811],{"className":53812,"style":322},[154],[19,53814,49],{"className":53815},[75,76],[19,53817,53818,53821],{"style":387},[19,53819],{"className":53820,"style":322},[154],[19,53822,53824],{"className":53823,"style":29053},[29052],[19,53825,28986],{"className":53826},[75],[19,53828,173],{"className":53829},[172],[19,53831,53833],{"className":53832},[142],[19,53834,53836],{"className":53835,"style":5671},[146],[19,53837],{},[19,53839],{"className":53840},[88,422],[19,53842,53844],{"className":53843,"style":16031},[88,16030],[19,53845,26668],{"className":53846},[16035,44882],[15,53848,53849],{},"is monotonically bounded. Thus, by the Abel's test, we know that",[19,53851,53853],{"className":53852,"translate":23},[187],[19,53854,53856,53904],{"className":53855,"translate":23},[22],[19,53857,53859],{"className":53858},[27],[29,53860,53861],{"xmlns":31,"display":196},[33,53862,53863,53901],{},[36,53864,53865,53879],{},[26840,53866,53867,53869,53877],{},[43,53868,26844],{},[36,53870,53871,53873,53875],{},[39,53872,3297],{},[43,53874,267],{},[108,53876,3495],{},[39,53878,26855],{"mathvariant":275},[36,53880,53881,53883,53889,53899],{},[43,53882,29161],{"fence":62},[103,53884,53885,53887],{},[39,53886,3484],{},[39,53888,3297],{},[269,53890,53891,53897],{},[28980,53892,53893,53895],{"accent":62},[39,53894,49],{},[43,53896,28986],{},[39,53898,3297],{},[43,53900,29161],{"fence":62},[54,53902,53903],{"encoding":56},"\\sum_{n=1}^{\\infty}{\\left| a_{n}\\bar{x}^{n} \\right|}",[19,53905,53907],{"className":53906,"ariaHidden":62},[61],[19,53908,53910,53913,53980,53983],{"className":53909},[66],[19,53911],{"className":53912,"style":26984},[70],[19,53914,53916],{"className":53915},[304,305],[19,53917,53919,53972],{"className":53918},[137,138],[19,53920,53922,53969],{"className":53921},[142],[19,53923,53925,53945,53955],{"className":53924,"style":26997},[146],[19,53926,53927,53930],{"style":27000},[19,53928],{"className":53929,"style":27004},[154],[19,53931,53933],{"className":53932},[159,160,161,162],[19,53934,53936,53939,53942],{"className":53935},[75,162],[19,53937,3297],{"className":53938},[75,76,162],[19,53940,267],{"className":53941},[335,162],[19,53943,3495],{"className":53944},[75,162],[19,53946,53947,53950],{"style":27022},[19,53948],{"className":53949,"style":27004},[154],[19,53951,53952],{},[19,53953,26844],{"className":53954},[304,27031,27032],[19,53956,53957,53960],{"style":27035},[19,53958],{"className":53959,"style":27004},[154],[19,53961,53963],{"className":53962},[159,160,161,162],[19,53964,53966],{"className":53965},[75,162],[19,53967,26855],{"className":53968},[75,162],[19,53970,173],{"className":53971},[172],[19,53973,53975],{"className":53974},[142],[19,53976,53978],{"className":53977,"style":27057},[146],[19,53979],{},[19,53981],{"className":53982,"style":412},[411],[19,53984,53986],{"className":53985},[75],[19,53987,53989,53992,54035,54095],{"className":53988},[16026],[19,53990,29161],{"className":53991,"style":16031},[81,16030],[19,53993,53995,53998],{"className":53994},[75],[19,53996,3484],{"className":53997},[75,76],[19,53999,54001],{"className":54000},[133],[19,54002,54004,54027],{"className":54003},[137,138],[19,54005,54007,54024],{"className":54006},[142],[19,54008,54010],{"className":54009,"style":3609},[146],[19,54011,54012,54015],{"style":150},[19,54013],{"className":54014,"style":155},[154],[19,54016,54018],{"className":54017},[159,160,161,162],[19,54019,54021],{"className":54020},[75,162],[19,54022,3297],{"className":54023},[75,76,162],[19,54025,173],{"className":54026},[172],[19,54028,54030],{"className":54029},[142],[19,54031,54033],{"className":54032,"style":180},[146],[19,54034],{},[19,54036,54038,54069],{"className":54037},[75],[19,54039,54041],{"className":54040},[75,29025],[19,54042,54044],{"className":54043},[137],[19,54045,54047],{"className":54046},[142],[19,54048,54050,54058],{"className":54049,"style":29035},[146],[19,54051,54052,54055],{"style":387},[19,54053],{"className":54054,"style":322},[154],[19,54056,49],{"className":54057},[75,76],[19,54059,54060,54063],{"style":387},[19,54061],{"className":54062,"style":322},[154],[19,54064,54066],{"className":54065,"style":29053},[29052],[19,54067,28986],{"className":54068},[75],[19,54070,54072],{"className":54071},[133],[19,54073,54075],{"className":54074},[137],[19,54076,54078],{"className":54077},[142],[19,54079,54081],{"className":54080,"style":4148},[146],[19,54082,54083,54086],{"style":648},[19,54084],{"className":54085,"style":155},[154],[19,54087,54089],{"className":54088},[159,160,161,162],[19,54090,54092],{"className":54091},[75,162],[19,54093,3297],{"className":54094},[75,76,162],[19,54096,29161],{"className":54097,"style":16031},[88,16030],[15,54099,54100,54101,54139,54140,54190],{},"converges. This contradicts the assumption. Therefore, ",[19,54102,54104,54121],{"className":54103,"translate":23},[22],[19,54105,54107],{"className":54106},[27],[29,54108,54109],{"xmlns":31},[33,54110,54111,54119],{},[36,54112,54113,54115,54117],{},[43,54114,46],{"stretchy":45},[108,54116,3518],{},[43,54118,52],{"stretchy":45},[54,54120,51402],{"encoding":56},[19,54122,54124],{"className":54123,"ariaHidden":62},[61],[19,54125,54127,54130,54133,54136],{"className":54126},[66],[19,54128],{"className":54129,"style":71},[70],[19,54131,46],{"className":54132},[81],[19,54134,3518],{"className":54135},[75],[19,54137,52],{"className":54138},[88]," also converges only at ",[19,54141,54143,54160],{"className":54142,"translate":23},[22],[19,54144,54146],{"className":54145},[27],[29,54147,54148],{"xmlns":31},[33,54149,54150,54158],{},[36,54151,54152,54154,54156],{},[39,54153,49],{},[43,54155,267],{},[108,54157,110],{},[54,54159,51806],{"encoding":56},[19,54161,54163,54181],{"className":54162,"ariaHidden":62},[61],[19,54164,54166,54169,54172,54175,54178],{"className":54165},[66],[19,54167],{"className":54168,"style":3309},[70],[19,54170,49],{"className":54171},[75,76],[19,54173],{"className":54174,"style":613},[411],[19,54176,267],{"className":54177},[335],[19,54179],{"className":54180,"style":613},[411],[19,54182,54184,54187],{"className":54183},[66],[19,54185],{"className":54186,"style":1214},[70],[19,54188,110],{"className":54189},[75],", and the proposition holds.",[15,54192,54193,54194,54232,54233,54289,54290,54374,54375,54447,54448,54504,54505,13643,54533,54621,54622,7830],{},"Now let the convergence interval of power series ",[19,54195,54197,54214],{"className":54196,"translate":23},[22],[19,54198,54200],{"className":54199},[27],[29,54201,54202],{"xmlns":31},[33,54203,54204,54212],{},[36,54205,54206,54208,54210],{},[43,54207,46],{"stretchy":45},[108,54209,3495],{},[43,54211,52],{"stretchy":45},[54,54213,51363],{"encoding":56},[19,54215,54217],{"className":54216,"ariaHidden":62},[61],[19,54218,54220,54223,54226,54229],{"className":54219},[66],[19,54221],{"className":54222,"style":71},[70],[19,54224,46],{"className":54225},[81],[19,54227,3495],{"className":54228},[75],[19,54230,52],{"className":54231},[88]," be ",[19,54234,54236,54259],{"className":54235,"translate":23},[22],[19,54237,54239],{"className":54238},[27],[29,54240,54241],{"xmlns":31},[33,54242,54243,54257],{},[36,54244,54245,54247,54249,54251,54253,54255],{},[43,54246,46],{"stretchy":45},[43,54248,240],{},[39,54250,10758],{},[43,54252,7830],{"separator":62},[39,54254,10758],{},[43,54256,52],{"stretchy":45},[54,54258,33497],{"encoding":56},[19,54260,54262],{"className":54261,"ariaHidden":62},[61],[19,54263,54265,54268,54271,54274,54277,54280,54283,54286],{"className":54264},[66],[19,54266],{"className":54267,"style":71},[70],[19,54269,46],{"className":54270},[81],[19,54272,240],{"className":54273},[75],[19,54275,10758],{"className":54276,"style":10844},[75,76],[19,54278,7830],{"className":54279},[8036],[19,54281],{"className":54282,"style":412},[411],[19,54284,10758],{"className":54285,"style":10844},[75,76],[19,54287,52],{"className":54288},[88],", where ",[19,54291,54293,54311],{"className":54292,"translate":23},[22],[19,54294,54296],{"className":54295},[27],[29,54297,54298],{"xmlns":31},[33,54299,54300,54308],{},[36,54301,54302,54304,54306],{},[39,54303,10758],{},[43,54305,28989],{"mathvariant":275},[108,54307,110],{},[54,54309,54310],{"encoding":56},"R\\neq 0",[19,54312,54314,54365],{"className":54313,"ariaHidden":62},[61],[19,54315,54317,54320,54323,54326,54362],{"className":54316},[66],[19,54318],{"className":54319,"style":6202},[70],[19,54321,10758],{"className":54322,"style":10844},[75,76],[19,54324],{"className":54325,"style":613},[411],[19,54327,54329,54356,54359],{"className":54328},[335],[19,54330,54332],{"className":54331},[335],[19,54333,54335],{"className":54334},[75,29069],[19,54336,54338],{"className":54337},[29073],[19,54339,54341,54344,54353],{"className":54340},[29077],[19,54342],{"className":54343,"style":6202},[70],[19,54345,54347],{"className":54346},[29084],[19,54348,54350],{"className":54349},[75],[19,54351,29091],{"className":54352},[335],[19,54354],{"className":54355},[29095],[19,54357],{"className":54358},[411,29099],[19,54360,267],{"className":54361},[335],[19,54363],{"className":54364,"style":613},[411],[19,54366,54368,54371],{"className":54367},[66],[19,54369],{"className":54370,"style":1214},[70],[19,54372,110],{"className":54373},[75],". Let ",[19,54376,54378,54395],{"className":54377,"translate":23},[22],[19,54379,54381],{"className":54380},[27],[29,54382,54383],{"xmlns":31},[33,54384,54385,54393],{},[36,54386,54387],{},[103,54388,54389,54391],{},[39,54390,49],{},[108,54392,110],{},[54,54394,113],{"encoding":56},[19,54396,54398],{"className":54397,"ariaHidden":62},[61],[19,54399,54401,54404],{"className":54400},[66],[19,54402],{"className":54403,"style":123},[70],[19,54405,54407,54410],{"className":54406},[75],[19,54408,49],{"className":54409},[75,76],[19,54411,54413],{"className":54412},[133],[19,54414,54416,54439],{"className":54415},[137,138],[19,54417,54419,54436],{"className":54418},[142],[19,54420,54422],{"className":54421,"style":147},[146],[19,54423,54424,54427],{"style":150},[19,54425],{"className":54426,"style":155},[154],[19,54428,54430],{"className":54429},[159,160,161,162],[19,54431,54433],{"className":54432},[75,162],[19,54434,110],{"className":54435},[75,162],[19,54437,173],{"className":54438},[172],[19,54440,54442],{"className":54441},[142],[19,54443,54445],{"className":54444,"style":180},[146],[19,54446],{}," be any nonzero point in ",[19,54449,54451,54474],{"className":54450,"translate":23},[22],[19,54452,54454],{"className":54453},[27],[29,54455,54456],{"xmlns":31},[33,54457,54458,54472],{},[36,54459,54460,54462,54464,54466,54468,54470],{},[43,54461,46],{"stretchy":45},[43,54463,240],{},[39,54465,10758],{},[43,54467,7830],{"separator":62},[39,54469,10758],{},[43,54471,52],{"stretchy":45},[54,54473,33497],{"encoding":56},[19,54475,54477],{"className":54476,"ariaHidden":62},[61],[19,54478,54480,54483,54486,54489,54492,54495,54498,54501],{"className":54479},[66],[19,54481],{"className":54482,"style":71},[70],[19,54484,46],{"className":54485},[81],[19,54487,240],{"className":54488},[75],[19,54490,10758],{"className":54491,"style":10844},[75,76],[19,54493,7830],{"className":54494},[8036],[19,54496],{"className":54497,"style":412},[411],[19,54499,10758],{"className":54500,"style":10844},[75,76],[19,54502,52],{"className":54503},[88],". As shown in the proof of the Abel's theorem, there exist positive numbers ",[19,54506,54508,54521],{"className":54507,"translate":23},[22],[19,54509,54511],{"className":54510},[27],[29,54512,54513],{"xmlns":31},[33,54514,54515,54519],{},[36,54516,54517],{},[39,54518,29955],{},[54,54520,29955],{"encoding":56},[19,54522,54524],{"className":54523,"ariaHidden":62},[61],[19,54525,54527,54530],{"className":54526},[66],[19,54528],{"className":54529,"style":7738},[70],[19,54531,29955],{"className":54532,"style":13222},[75,76],[19,54534,54536,54564],{"className":54535,"translate":23},[22],[19,54537,54539],{"className":54538},[27],[29,54540,54541],{"xmlns":31},[33,54542,54543,54561],{},[36,54544,54545,54547,54549,54551,54553,54555,54557,54559],{},[39,54546,30769],{},[43,54548,46],{"stretchy":45},[108,54550,110],{},[43,54552,29168],{},[39,54554,30769],{},[43,54556,29168],{},[108,54558,3495],{},[43,54560,52],{"stretchy":45},[54,54562,54563],{"encoding":56},"r(0\u003Cr\u003C1)",[19,54565,54567,54591,54609],{"className":54566,"ariaHidden":62},[61],[19,54568,54570,54573,54576,54579,54582,54585,54588],{"className":54569},[66],[19,54571],{"className":54572,"style":71},[70],[19,54574,30769],{"className":54575,"style":31593},[75,76],[19,54577,46],{"className":54578},[81],[19,54580,110],{"className":54581},[75],[19,54583],{"className":54584,"style":613},[411],[19,54586,29168],{"className":54587},[335],[19,54589],{"className":54590,"style":613},[411],[19,54592,54594,54597,54600,54603,54606],{"className":54593},[66],[19,54595],{"className":54596,"style":13562},[70],[19,54598,30769],{"className":54599,"style":31593},[75,76],[19,54601],{"className":54602,"style":613},[411],[19,54604,29168],{"className":54605},[335],[19,54607],{"className":54608,"style":613},[411],[19,54610,54612,54615,54618],{"className":54611},[66],[19,54613],{"className":54614,"style":71},[70],[19,54616,3495],{"className":54617},[75],[19,54619,52],{"className":54620},[88]," such that for all positive integers 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a_{n}x_{0}^{n} 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the ratio test, the 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Therefore, by the comparison test, the series",[19,55812,55814],{"className":55813,"translate":23},[187],[19,55815,55817,55871],{"className":55816,"translate":23},[22],[19,55818,55820],{"className":55819},[27],[29,55821,55822],{"xmlns":31,"display":196},[33,55823,55824,55868],{},[36,55825,55826,55840],{},[26840,55827,55828,55830,55838],{},[43,55829,26844],{},[36,55831,55832,55834,55836],{},[39,55833,3297],{},[43,55835,267],{},[108,55837,110],{},[39,55839,26855],{"mathvariant":275},[36,55841,55842,55844,55846,55852,55866],{},[43,55843,29161],{"fence":62},[39,55845,3297],{},[103,55847,55848,55850],{},[39,55849,3484],{},[39,55851,3297],{},[4285,55853,55854,55856,55858],{},[39,55855,49],{},[108,55857,110],{},[36,55859,55860,55862,55864],{},[39,55861,3297],{},[43,55863,240],{},[108,55865,3495],{},[43,55867,29161],{"fence":62},[54,55869,55870],{"encoding":56},"\\sum_{n=0}^{\\infty}{\\left| na_{n}x_{0}^{n-1} \\right|}",[19,55872,55874],{"className":55873,"ariaHidden":62},[61],[19,55875,55877,55880,55947,55950],{"className":55876},[66],[19,55878],{"className":55879,"style":26984},[70],[19,55881,55883],{"className":55882},[304,305],[19,55884,55886,55939],{"className":55885},[137,138],[19,55887,55889,55936],{"className":55888},[142],[19,55890,55892,55912,55922],{"className":55891,"style":26997},[146],[19,55893,55894,55897],{"style":27000},[19,55895],{"className":55896,"style":27004},[154],[19,55898,55900],{"className":55899},[159,160,161,162],[19,55901,55903,55906,55909],{"className":55902},[75,162],[19,55904,3297],{"className":55905},[75,76,162],[19,55907,267],{"className":55908},[335,162],[19,55910,110],{"className":55911},[75,162],[19,55913,55914,55917],{"style":27022},[19,55915],{"className":55916,"style":27004},[154],[19,55918,55919],{},[19,55920,26844],{"className":55921},[304,27031,27032],[19,55923,55924,55927],{"style":27035},[19,55925],{"className":55926,"style":27004},[154],[19,55928,55930],{"className":55929},[159,160,161,162],[19,55931,55933],{"className":55932},[75,162],[19,55934,26855],{"className":55935},[75,162],[19,55937,173],{"className":55938},[172],[19,55940,55942],{"className":55941},[142],[19,55943,55945],{"className":55944,"style":27057},[146],[19,55946],{},[19,55948],{"className":55949,"style":412},[411],[19,55951,55953],{"className":55952},[75],[19,55954,55956,55993,55996,56039,56102],{"className":55955},[16026],[19,55957,55959],{"className":55958},[81],[19,55960,55962],{"className":55961},[16035,30439],[19,55963,55965,55985],{"className":55964},[137,138],[19,55966,55968,55982],{"className":55967},[142],[19,55969,55971],{"className":55970,"style":30449},[146],[19,55972,55973,55976],{"style":30452},[19,55974],{"className":55975,"style":30456},[154],[19,55977,55978],{"style":30459},[30461,55979,55980],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,55981],{"d":30470},[19,55983,173],{"className":55984},[172],[19,55986,55988],{"className":55987},[142],[19,55989,55991],{"className":55990,"style":30480},[146],[19,55992],{},[19,55994,3297],{"className":55995},[75,76],[19,55997,55999,56002],{"className":55998},[75],[19,56000,3484],{"className":56001},[75,76],[19,56003,56005],{"className":56004},[133],[19,56006,56008,56031],{"className":56007},[137,138],[19,56009,56011,56028],{"className":56010},[142],[19,56012,56014],{"className":56013,"style":3609},[146],[19,56015,56016,56019],{"style":150},[19,56017],{"className":56018,"style":155},[154],[19,56020,56022],{"className":56021},[159,160,161,162],[19,56023,56025],{"className":56024},[75,162],[19,56026,3297],{"className":56027},[75,76,162],[19,56029,173],{"className":56030},[172],[19,56032,56034],{"className":56033},[142],[19,56035,56037],{"className":56036,"style":180},[146],[19,56038],{},[19,56040,56042,56045],{"className":56041},[75],[19,56043,49],{"className":56044},[75,76],[19,56046,56048],{"className":56047},[133],[19,56049,56051,56094],{"className":56050},[137,138],[19,56052,56054,56091],{"className":56053},[142],[19,56055,56057,56071],{"className":56056,"style":3970},[146],[19,56058,56059,56062],{"style":55095},[19,56060],{"className":56061,"style":155},[154],[19,56063,56065],{"className":56064},[159,160,161,162],[19,56066,56068],{"className":56067},[75,162],[19,56069,110],{"className":56070},[75,162],[19,56072,56073,56076],{"style":648},[19,56074],{"className":56075,"style":155},[154],[19,56077,56079],{"className":56078},[159,160,161,162],[19,56080,56082,56085,56088],{"className":56081},[75,162],[19,56083,3297],{"className":56084},[75,76,162],[19,56086,240],{"className":56087},[455,162],[19,56089,3495],{"className":56090},[75,162],[19,56092,173],{"className":56093},[172],[19,56095,56097],{"className":56096},[142],[19,56098,56100],{"className":56099,"style":55137},[146],[19,56101],{},[19,56103,56105],{"className":56104},[88],[19,56106,56108],{"className":56107},[16035,30439],[19,56109,56111,56131],{"className":56110},[137,138],[19,56112,56114,56128],{"className":56113},[142],[19,56115,56117],{"className":56116,"style":30449},[146],[19,56118,56119,56122],{"style":30452},[19,56120],{"className":56121,"style":30456},[154],[19,56123,56124],{"style":30459},[30461,56125,56126],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,56127],{"d":30470},[19,56129,173],{"className":56130},[172],[19,56132,56134],{"className":56133},[142],[19,56135,56137],{"className":56136,"style":30480},[146],[19,56138],{},[15,56140,56141,56142,56180,56181,56253,56254,56326,56327,56383,56384,56422,56423,3447],{},"converges. This implies that power series ",[19,56143,56145,56162],{"className":56144,"translate":23},[22],[19,56146,56148],{"className":56147},[27],[29,56149,56150],{"xmlns":31},[33,56151,56152,56160],{},[36,56153,56154,56156,56158],{},[43,56155,46],{"stretchy":45},[108,56157,3518],{},[43,56159,52],{"stretchy":45},[54,56161,51402],{"encoding":56},[19,56163,56165],{"className":56164,"ariaHidden":62},[61],[19,56166,56168,56171,56174,56177],{"className":56167},[66],[19,56169],{"className":56170,"style":71},[70],[19,56172,46],{"className":56173},[81],[19,56175,3518],{"className":56176},[75],[19,56178,52],{"className":56179},[88]," is absolutely convergent (and hence convergent) at point ",[19,56182,56184,56201],{"className":56183,"translate":23},[22],[19,56185,56187],{"className":56186},[27],[29,56188,56189],{"xmlns":31},[33,56190,56191,56199],{},[36,56192,56193],{},[103,56194,56195,56197],{},[39,56196,49],{},[108,56198,110],{},[54,56200,113],{"encoding":56},[19,56202,56204],{"className":56203,"ariaHidden":62},[61],[19,56205,56207,56210],{"className":56206},[66],[19,56208],{"className":56209,"style":123},[70],[19,56211,56213,56216],{"className":56212},[75],[19,56214,49],{"className":56215},[75,76],[19,56217,56219],{"className":56218},[133],[19,56220,56222,56245],{"className":56221},[137,138],[19,56223,56225,56242],{"className":56224},[142],[19,56226,56228],{"className":56227,"style":147},[146],[19,56229,56230,56233],{"style":150},[19,56231],{"className":56232,"style":155},[154],[19,56234,56236],{"className":56235},[159,160,161,162],[19,56237,56239],{"className":56238},[75,162],[19,56240,110],{"className":56241},[75,162],[19,56243,173],{"className":56244},[172],[19,56246,56248],{"className":56247},[142],[19,56249,56251],{"className":56250,"style":180},[146],[19,56252],{},", as it converges absolutely. Since ",[19,56255,56257,56274],{"className":56256,"translate":23},[22],[19,56258,56260],{"className":56259},[27],[29,56261,56262],{"xmlns":31},[33,56263,56264,56272],{},[36,56265,56266],{},[103,56267,56268,56270],{},[39,56269,49],{},[108,56271,110],{},[54,56273,113],{"encoding":56},[19,56275,56277],{"className":56276,"ariaHidden":62},[61],[19,56278,56280,56283],{"className":56279},[66],[19,56281],{"className":56282,"style":123},[70],[19,56284,56286,56289],{"className":56285},[75],[19,56287,49],{"className":56288},[75,76],[19,56290,56292],{"className":56291},[133],[19,56293,56295,56318],{"className":56294},[137,138],[19,56296,56298,56315],{"className":56297},[142],[19,56299,56301],{"className":56300,"style":147},[146],[19,56302,56303,56306],{"style":150},[19,56304],{"className":56305,"style":155},[154],[19,56307,56309],{"className":56308},[159,160,161,162],[19,56310,56312],{"className":56311},[75,162],[19,56313,110],{"className":56314},[75,162],[19,56316,173],{"className":56317},[172],[19,56319,56321],{"className":56320},[142],[19,56322,56324],{"className":56323,"style":180},[146],[19,56325],{}," is an arbitrary point in ",[19,56328,56330,56353],{"className":56329,"translate":23},[22],[19,56331,56333],{"className":56332},[27],[29,56334,56335],{"xmlns":31},[33,56336,56337,56351],{},[36,56338,56339,56341,56343,56345,56347,56349],{},[43,56340,46],{"stretchy":45},[43,56342,240],{},[39,56344,10758],{},[43,56346,7830],{"separator":62},[39,56348,10758],{},[43,56350,52],{"stretchy":45},[54,56352,33497],{"encoding":56},[19,56354,56356],{"className":56355,"ariaHidden":62},[61],[19,56357,56359,56362,56365,56368,56371,56374,56377,56380],{"className":56358},[66],[19,56360],{"className":56361,"style":71},[70],[19,56363,46],{"className":56364},[81],[19,56366,240],{"className":56367},[75],[19,56369,10758],{"className":56370,"style":10844},[75,76],[19,56372,7830],{"className":56373},[8036],[19,56375],{"className":56376,"style":412},[411],[19,56378,10758],{"className":56379,"style":10844},[75,76],[19,56381,52],{"className":56382},[88],", power series ",[19,56385,56387,56404],{"className":56386,"translate":23},[22],[19,56388,56390],{"className":56389},[27],[29,56391,56392],{"xmlns":31},[33,56393,56394,56402],{},[36,56395,56396,56398,56400],{},[43,56397,46],{"stretchy":45},[108,56399,3518],{},[43,56401,52],{"stretchy":45},[54,56403,51402],{"encoding":56},[19,56405,56407],{"className":56406,"ariaHidden":62},[61],[19,56408,56410,56413,56416,56419],{"className":56409},[66],[19,56411],{"className":56412,"style":71},[70],[19,56414,46],{"className":56415},[81],[19,56417,3518],{"className":56418},[75],[19,56420,52],{"className":56421},[88]," converges on the interval ",[19,56424,56426,56449],{"className":56425,"translate":23},[22],[19,56427,56429],{"className":56428},[27],[29,56430,56431],{"xmlns":31},[33,56432,56433,56447],{},[36,56434,56435,56437,56439,56441,56443,56445],{},[43,56436,46],{"stretchy":45},[43,56438,240],{},[39,56440,10758],{},[43,56442,7830],{"separator":62},[39,56444,10758],{},[43,56446,52],{"stretchy":45},[54,56448,33497],{"encoding":56},[19,56450,56452],{"className":56451,"ariaHidden":62},[61],[19,56453,56455,56458,56461,56464,56467,56470,56473,56476],{"className":56454},[66],[19,56456],{"className":56457,"style":71},[70],[19,56459,46],{"className":56460},[81],[19,56462,240],{"className":56463},[75],[19,56465,10758],{"className":56466,"style":10844},[75,76],[19,56468,7830],{"className":56469},[8036],[19,56471],{"className":56472,"style":412},[411],[19,56474,10758],{"className":56475,"style":10844},[75,76],[19,56477,52],{"className":56478},[88],[15,56480,56481,56482,56520,56521,56549,56550,56616,56617,56655,56656,56825,56826,32562,56886,57065,57066,57104,57105,57187,57188,42595],{},"Next, we need to prove that power series ",[19,56483,56485,56502],{"className":56484,"translate":23},[22],[19,56486,56488],{"className":56487},[27],[29,56489,56490],{"xmlns":31},[33,56491,56492,56500],{},[36,56493,56494,56496,56498],{},[43,56495,46],{"stretchy":45},[108,56497,3518],{},[43,56499,52],{"stretchy":45},[54,56501,51402],{"encoding":56},[19,56503,56505],{"className":56504,"ariaHidden":62},[61],[19,56506,56508,56511,56514,56517],{"className":56507},[66],[19,56509],{"className":56510,"style":71},[70],[19,56512,46],{"className":56513},[81],[19,56515,3518],{"className":56516},[75],[19,56518,52],{"className":56519},[88]," does not converge for all ",[19,56522,56524,56537],{"className":56523,"translate":23},[22],[19,56525,56527],{"className":56526},[27],[29,56528,56529],{"xmlns":31},[33,56530,56531,56535],{},[36,56532,56533],{},[39,56534,49],{},[54,56536,49],{"encoding":56},[19,56538,56540],{"className":56539,"ariaHidden":62},[61],[19,56541,56543,56546],{"className":56542},[66],[19,56544],{"className":56545,"style":3309},[70],[19,56547,49],{"className":56548},[75,76]," satisfying ",[19,56551,56553,56577],{"className":56552,"translate":23},[22],[19,56554,56556],{"className":56555},[27],[29,56557,56558],{"xmlns":31},[33,56559,56560,56574],{},[36,56561,56562,56570,56572],{},[36,56563,56564,56566,56568],{},[43,56565,29161],{"fence":62},[39,56567,49],{},[43,56569,29161],{"fence":62},[43,56571,29448],{},[39,56573,10758],{},[54,56575,56576],{"encoding":56},"\\left| x \\right|>R",[19,56578,56580,56607],{"className":56579,"ariaHidden":62},[61],[19,56581,56583,56586,56598,56601,56604],{"className":56582},[66],[19,56584],{"className":56585,"style":71},[70],[19,56587,56589,56592,56595],{"className":56588},[16026],[19,56590,29161],{"className":56591,"style":16031},[81,16030],[19,56593,49],{"className":56594},[75,76],[19,56596,29161],{"className":56597,"style":16031},[88,16030],[19,56599],{"className":56600,"style":613},[411],[19,56602,29448],{"className":56603},[335],[19,56605],{"className":56606,"style":613},[411],[19,56608,56610,56613],{"className":56609},[66],[19,56611],{"className":56612,"style":7738},[70],[19,56614,10758],{"className":56615,"style":10844},[75,76],". Suppose ",[19,56618,56620,56637],{"className":56619,"translate":23},[22],[19,56621,56623],{"className":56622},[27],[29,56624,56625],{"xmlns":31},[33,56626,56627,56635],{},[36,56628,56629,56631,56633],{},[43,56630,46],{"stretchy":45},[108,56632,3518],{},[43,56634,52],{"stretchy":45},[54,56636,51402],{"encoding":56},[19,56638,56640],{"className":56639,"ariaHidden":62},[61],[19,56641,56643,56646,56649,56652],{"className":56642},[66],[19,56644],{"className":56645,"style":71},[70],[19,56647,46],{"className":56648},[81],[19,56650,3518],{"className":56651},[75],[19,56653,52],{"className":56654},[88]," converges at a point ",[19,56657,56659,56697],{"className":56658,"translate":23},[22],[19,56660,56662],{"className":56661},[27],[29,56663,56664],{"xmlns":31},[33,56665,56666,56694],{},[36,56667,56668,56674,56676,56688,56690,56692],{},[103,56669,56670,56672],{},[39,56671,49],{},[108,56673,110],{},[43,56675,46],{"stretchy":45},[36,56677,56678,56680,56686],{},[43,56679,29161],{"fence":62},[103,56681,56682,56684],{},[39,56683,49],{},[108,56685,110],{},[43,56687,29161],{"fence":62},[43,56689,29448],{},[39,56691,10758],{},[43,56693,52],{"stretchy":45},[54,56695,56696],{"encoding":56},"x_{0} (\\left| x_{0} \\right|>R)",[19,56698,56700,56813],{"className":56699,"ariaHidden":62},[61],[19,56701,56703,56706,56749,56752,56804,56807,56810],{"className":56702},[66],[19,56704],{"className":56705,"style":71},[70],[19,56707,56709,56712],{"className":56708},[75],[19,56710,49],{"className":56711},[75,76],[19,56713,56715],{"className":56714},[133],[19,56716,56718,56741],{"className":56717},[137,138],[19,56719,56721,56738],{"className":56720},[142],[19,56722,56724],{"className":56723,"style":147},[146],[19,56725,56726,56729],{"style":150},[19,56727],{"className":56728,"style":155},[154],[19,56730,56732],{"className":56731},[159,160,161,162],[19,56733,56735],{"className":56734},[75,162],[19,56736,110],{"className":56737},[75,162],[19,56739,173],{"className":56740},[172],[19,56742,56744],{"className":56743},[142],[19,56745,56747],{"className":56746,"style":180},[146],[19,56748],{},[19,56750,46],{"className":56751},[81],[19,56753,56755,56758,56801],{"className":56754},[16026],[19,56756,29161],{"className":56757,"style":16031},[81,16030],[19,56759,56761,56764],{"className":56760},[75],[19,56762,49],{"className":56763},[75,76],[19,56765,56767],{"className":56766},[133],[19,56768,56770,56793],{"className":56769},[137,138],[19,56771,56773,56790],{"className":56772},[142],[19,56774,56776],{"className":56775,"style":147},[146],[19,56777,56778,56781],{"style":150},[19,56779],{"className":56780,"style":155},[154],[19,56782,56784],{"className":56783},[159,160,161,162],[19,56785,56787],{"className":56786},[75,162],[19,56788,110],{"className":56789},[75,162],[19,56791,173],{"className":56792},[172],[19,56794,56796],{"className":56795},[142],[19,56797,56799],{"className":56798,"style":180},[146],[19,56800],{},[19,56802,29161],{"className":56803,"style":16031},[88,16030],[19,56805],{"className":56806,"style":613},[411],[19,56808,29448],{"className":56809},[335],[19,56811],{"className":56812,"style":613},[411],[19,56814,56816,56819,56822],{"className":56815},[66],[19,56817],{"className":56818,"style":71},[70],[19,56820,10758],{"className":56821,"style":10844},[75,76],[19,56823,52],{"className":56824},[88],", then there exists a number ",[19,56827,56829,56846],{"className":56828,"translate":23},[22],[19,56830,56832],{"className":56831},[27],[29,56833,56834],{"xmlns":31},[33,56835,56836,56844],{},[36,56837,56838],{},[28980,56839,56840,56842],{"accent":62},[39,56841,49],{},[43,56843,28986],{},[54,56845,42900],{"encoding":56},[19,56847,56849],{"className":56848,"ariaHidden":62},[61],[19,56850,56852,56855],{"className":56851},[66],[19,56853],{"className":56854,"style":29035},[70],[19,56856,56858],{"className":56857},[75,29025],[19,56859,56861],{"className":56860},[137],[19,56862,56864],{"className":56863},[142],[19,56865,56867,56875],{"className":56866,"style":29035},[146],[19,56868,56869,56872],{"style":387},[19,56870],{"className":56871,"style":322},[154],[19,56873,49],{"className":56874},[75,76],[19,56876,56877,56880],{"style":387},[19,56878],{"className":56879,"style":322},[154],[19,56881,56883],{"className":56882,"style":29053},[29052],[19,56884,28986],{"className":56885},[75],[19,56887,56889,56931],{"className":56888,"translate":23},[22],[19,56890,56892],{"className":56891},[27],[29,56893,56894],{"xmlns":31},[33,56895,56896,56928],{},[36,56897,56898,56910,56912,56924,56926],{},[36,56899,56900,56902,56908],{},[43,56901,29161],{"fence":62},[103,56903,56904,56906],{},[39,56905,49],{},[108,56907,110],{},[43,56909,29161],{"fence":62},[43,56911,29448],{},[36,56913,56914,56916,56922],{},[43,56915,29161],{"fence":62},[28980,56917,56918,56920],{"accent":62},[39,56919,49],{},[43,56921,28986],{},[43,56923,29161],{"fence":62},[43,56925,29448],{},[39,56927,10758],{},[54,56929,56930],{"encoding":56},"\\left| x_{0} \\right|>\\left| \\bar{x} \\right|>R",[19,56932,56934,57001,57056],{"className":56933,"ariaHidden":62},[61],[19,56935,56937,56940,56992,56995,56998],{"className":56936},[66],[19,56938],{"className":56939,"style":71},[70],[19,56941,56943,56946,56989],{"className":56942},[16026],[19,56944,29161],{"className":56945,"style":16031},[81,16030],[19,56947,56949,56952],{"className":56948},[75],[19,56950,49],{"className":56951},[75,76],[19,56953,56955],{"className":56954},[133],[19,56956,56958,56981],{"className":56957},[137,138],[19,56959,56961,56978],{"className":56960},[142],[19,56962,56964],{"className":56963,"style":147},[146],[19,56965,56966,56969],{"style":150},[19,56967],{"className":56968,"style":155},[154],[19,56970,56972],{"className":56971},[159,160,161,162],[19,56973,56975],{"className":56974},[75,162],[19,56976,110],{"className":56977},[75,162],[19,56979,173],{"className":56980},[172],[19,56982,56984],{"className":56983},[142],[19,56985,56987],{"className":56986,"style":180},[146],[19,56988],{},[19,56990,29161],{"className":56991,"style":16031},[88,16030],[19,56993],{"className":56994,"style":613},[411],[19,56996,29448],{"className":56997},[335],[19,56999],{"className":57000,"style":613},[411],[19,57002,57004,57007,57047,57050,57053],{"className":57003},[66],[19,57005],{"className":57006,"style":71},[70],[19,57008,57010,57013,57044],{"className":57009},[16026],[19,57011,29161],{"className":57012,"style":16031},[81,16030],[19,57014,57016],{"className":57015},[75,29025],[19,57017,57019],{"className":57018},[137],[19,57020,57022],{"className":57021},[142],[19,57023,57025,57033],{"className":57024,"style":29035},[146],[19,57026,57027,57030],{"style":387},[19,57028],{"className":57029,"style":322},[154],[19,57031,49],{"className":57032},[75,76],[19,57034,57035,57038],{"style":387},[19,57036],{"className":57037,"style":322},[154],[19,57039,57041],{"className":57040,"style":29053},[29052],[19,57042,28986],{"className":57043},[75],[19,57045,29161],{"className":57046,"style":16031},[88,16030],[19,57048],{"className":57049,"style":613},[411],[19,57051,29448],{"className":57052},[335],[19,57054],{"className":57055,"style":613},[411],[19,57057,57059,57062],{"className":57058},[66],[19,57060],{"className":57061,"style":7738},[70],[19,57063,10758],{"className":57064,"style":10844},[75,76],". By the Abel's theorem, power series ",[19,57067,57069,57086],{"className":57068,"translate":23},[22],[19,57070,57072],{"className":57071},[27],[29,57073,57074],{"xmlns":31},[33,57075,57076,57084],{},[36,57077,57078,57080,57082],{},[43,57079,46],{"stretchy":45},[108,57081,3518],{},[43,57083,52],{"stretchy":45},[54,57085,51402],{"encoding":56},[19,57087,57089],{"className":57088,"ariaHidden":62},[61],[19,57090,57092,57095,57098,57101],{"className":57091},[66],[19,57093],{"className":57094,"style":71},[70],[19,57096,46],{"className":57097},[81],[19,57099,3518],{"className":57100},[75],[19,57102,52],{"className":57103},[88]," converges absolutely at ",[19,57106,57108,57129],{"className":57107,"translate":23},[22],[19,57109,57111],{"className":57110},[27],[29,57112,57113],{"xmlns":31},[33,57114,57115,57127],{},[36,57116,57117,57119,57121],{},[39,57118,49],{},[43,57120,267],{},[28980,57122,57123,57125],{"accent":62},[39,57124,49],{},[43,57126,28986],{},[54,57128,32971],{"encoding":56},[19,57130,57132,57150],{"className":57131,"ariaHidden":62},[61],[19,57133,57135,57138,57141,57144,57147],{"className":57134},[66],[19,57136],{"className":57137,"style":3309},[70],[19,57139,49],{"className":57140},[75,76],[19,57142],{"className":57143,"style":613},[411],[19,57145,267],{"className":57146},[335],[19,57148],{"className":57149,"style":613},[411],[19,57151,57153,57156],{"className":57152},[66],[19,57154],{"className":57155,"style":29035},[70],[19,57157,57159],{"className":57158},[75,29025],[19,57160,57162],{"className":57161},[137],[19,57163,57165],{"className":57164},[142],[19,57166,57168,57176],{"className":57167,"style":29035},[146],[19,57169,57170,57173],{"style":387},[19,57171],{"className":57172,"style":322},[154],[19,57174,49],{"className":57175},[75,76],[19,57177,57178,57181],{"style":387},[19,57179],{"className":57180,"style":322},[154],[19,57182,57184],{"className":57183,"style":29053},[29052],[19,57185,28986],{"className":57186},[75],". However, when ",[19,57189,57191,57219],{"className":57190,"translate":23},[22],[19,57192,57194],{"className":57193},[27],[29,57195,57196],{"xmlns":31},[33,57197,57198,57216],{},[36,57199,57200,57202,57204],{},[39,57201,3297],{},[43,57203,45059],{},[36,57205,57206,57208,57214],{},[43,57207,29161],{"fence":62},[28980,57209,57210,57212],{"accent":62},[39,57211,49],{},[43,57213,28986],{},[43,57215,29161],{"fence":62},[54,57217,57218],{"encoding":56},"n\\geq\\left| \\bar{x} \\right|",[19,57220,57222,57240],{"className":57221,"ariaHidden":62},[61],[19,57223,57225,57228,57231,57234,57237],{"className":57224},[66],[19,57226],{"className":57227,"style":45339},[70],[19,57229,3297],{"className":57230},[75,76],[19,57232],{"className":57233,"style":613},[411],[19,57235,45059],{"className":57236},[335],[19,57238],{"className":57239,"style":613},[411],[19,57241,57243,57246],{"className":57242},[66],[19,57244],{"className":57245,"style":71},[70],[19,57247,57249,57252,57283],{"className":57248},[16026],[19,57250,29161],{"className":57251,"style":16031},[81,16030],[19,57253,57255],{"className":57254},[75,29025],[19,57256,57258],{"className":57257},[137],[19,57259,57261],{"className":57260},[142],[19,57262,57264,57272],{"className":57263,"style":29035},[146],[19,57265,57266,57269],{"style":387},[19,57267],{"className":57268,"style":322},[154],[19,57270,49],{"className":57271},[75,76],[19,57273,57274,57277],{"style":387},[19,57275],{"className":57276,"style":322},[154],[19,57278,57280],{"className":57279,"style":29053},[29052],[19,57281,28986],{"className":57282},[75],[19,57284,29161],{"className":57285,"style":16031},[88,16030],[19,57287,57289],{"className":57288,"translate":23},[187],[19,57290,57292,57398],{"className":57291,"translate":23},[22],[19,57293,57295],{"className":57294},[27],[29,57296,57297],{"xmlns":31,"display":196},[33,57298,57299,57395],{},[36,57300,57301,57331,57333,57349,57371,57373],{},[36,57302,57303,57305,57307,57313,57329],{},[43,57304,29161],{"fence":62},[39,57306,3297],{},[103,57308,57309,57311],{},[39,57310,3484],{},[39,57312,3297],{},[269,57314,57315,57321],{},[28980,57316,57317,57319],{"accent":62},[39,57318,49],{},[43,57320,28986],{},[36,57322,57323,57325,57327],{},[39,57324,3297],{},[43,57326,240],{},[108,57328,3495],{},[43,57330,29161],{"fence":62},[43,57332,267],{},[226,57334,57335,57337],{},[39,57336,3297],{},[36,57338,57339,57341,57347],{},[43,57340,29161],{"fence":62},[28980,57342,57343,57345],{"accent":62},[39,57344,49],{},[43,57346,28986],{},[43,57348,29161],{"fence":62},[36,57350,57351,57353,57359,57369],{},[43,57352,29161],{"fence":62},[103,57354,57355,57357],{},[39,57356,3484],{},[39,57358,3297],{},[269,57360,57361,57367],{},[28980,57362,57363,57365],{"accent":62},[39,57364,49],{},[43,57366,28986],{},[39,57368,3297],{},[43,57370,29161],{"fence":62},[43,57372,45059],{},[36,57374,57375,57377,57383,57393],{},[43,57376,29161],{"fence":62},[103,57378,57379,57381],{},[39,57380,3484],{},[39,57382,3297],{},[269,57384,57385,57391],{},[28980,57386,57387,57389],{"accent":62},[39,57388,49],{},[43,57390,28986],{},[39,57392,3297],{},[43,57394,29161],{"fence":62},[54,57396,57397],{"encoding":56},"\\left| na_{n}\\bar{x}^{n-1} \\right|=\\frac{n}{\\left| \\bar{x} \\right|}\\left| a_{n}\\bar{x}^{n} \\right|\\geq \\left| a_{n}\\bar{x}^{n} \\right|",[19,57399,57401,57605,57835],{"className":57400,"ariaHidden":62},[61],[19,57402,57404,57407,57596,57599,57602],{"className":57403},[66],[19,57405],{"className":57406,"style":54988},[70],[19,57408,57410,57447,57450,57493,57559],{"className":57409},[16026],[19,57411,57413],{"className":57412},[81],[19,57414,57416],{"className":57415},[16035,30439],[19,57417,57419,57439],{"className":57418},[137,138],[19,57420,57422,57436],{"className":57421},[142],[19,57423,57425],{"className":57424,"style":30449},[146],[19,57426,57427,57430],{"style":30452},[19,57428],{"className":57429,"style":30456},[154],[19,57431,57432],{"style":30459},[30461,57433,57434],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,57435],{"d":30470},[19,57437,173],{"className":57438},[172],[19,57440,57442],{"className":57441},[142],[19,57443,57445],{"className":57444,"style":30480},[146],[19,57446],{},[19,57448,3297],{"className":57449},[75,76],[19,57451,57453,57456],{"className":57452},[75],[19,57454,3484],{"className":57455},[75,76],[19,57457,57459],{"className":57458},[133],[19,57460,57462,57485],{"className":57461},[137,138],[19,57463,57465,57482],{"className":57464},[142],[19,57466,57468],{"className":57467,"style":3609},[146],[19,57469,57470,57473],{"style":150},[19,57471],{"className":57472,"style":155},[154],[19,57474,57476],{"className":57475},[159,160,161,162],[19,57477,57479],{"className":57478},[75,162],[19,57480,3297],{"className":57481},[75,76,162],[19,57483,173],{"className":57484},[172],[19,57486,57488],{"className":57487},[142],[19,57489,57491],{"className":57490,"style":180},[146],[19,57492],{},[19,57494,57496,57527],{"className":57495},[75],[19,57497,57499],{"className":57498},[75,29025],[19,57500,57502],{"className":57501},[137],[19,57503,57505],{"className":57504},[142],[19,57506,57508,57516],{"className":57507,"style":29035},[146],[19,57509,57510,57513],{"style":387},[19,57511],{"className":57512,"style":322},[154],[19,57514,49],{"className":57515},[75,76],[19,57517,57518,57521],{"style":387},[19,57519],{"className":57520,"style":322},[154],[19,57522,57524],{"className":57523,"style":29053},[29052],[19,57525,28986],{"className":57526},[75],[19,57528,57530],{"className":57529},[133],[19,57531,57533],{"className":57532},[137],[19,57534,57536],{"className":57535},[142],[19,57537,57539],{"className":57538,"style":3970},[146],[19,57540,57541,57544],{"style":648},[19,57542],{"className":57543,"style":155},[154],[19,57545,57547],{"className":57546},[159,160,161,162],[19,57548,57550,57553,57556],{"className":57549},[75,162],[19,57551,3297],{"className":57552},[75,76,162],[19,57554,240],{"className":57555},[455,162],[19,57557,3495],{"className":57558},[75,162],[19,57560,57562],{"className":57561},[88],[19,57563,57565],{"className":57564},[16035,30439],[19,57566,57568,57588],{"className":57567},[137,138],[19,57569,57571,57585],{"className":57570},[142],[19,57572,57574],{"className":57573,"style":30449},[146],[19,57575,57576,57579],{"style":30452},[19,57577],{"className":57578,"style":30456},[154],[19,57580,57581],{"style":30459},[30461,57582,57583],{"xmlns":30463,"width":30464,"height":30465,"viewBox":30466},[30468,57584],{"d":30470},[19,57586,173],{"className":57587},[172],[19,57589,57591],{"className":57590},[142],[19,57592,57594],{"className":57593,"style":30480},[146],[19,57595],{},[19,57597],{"className":57598,"style":613},[411],[19,57600,267],{"className":57601},[335],[19,57603],{"className":57604,"style":613},[411],[19,57606,57608,57612,57711,57714,57826,57829,57832],{"className":57607},[66],[19,57609],{"className":57610,"style":57611},[70],"height:2.0436em;vertical-align:-0.936em;",[19,57613,57615,57618,57708],{"className":57614},[75],[19,57616],{"className":57617},[81,422],[19,57619,57621],{"className":57620},[226],[19,57622,57624,57700],{"className":57623},[137,138],[19,57625,57627,57697],{"className":57626},[142],[19,57628,57630,57678,57686],{"className":57629,"style":31431},[146],[19,57631,57632,57635],{"style":438},[19,57633],{"className":57634,"style":322},[154],[19,57636,57638],{"className":57637},[75],[19,57639,57641,57644,57675],{"className":57640},[16026],[19,57642,29161],{"className":57643,"style":16031},[81,16030],[19,57645,57647],{"className":57646},[75,29025],[19,57648,57650],{"className":57649},[137],[19,57651,57653],{"className":57652},[142],[19,57654,57656,57664],{"className":57655,"style":29035},[146],[19,57657,57658,57661],{"style":387},[19,57659],{"className":57660,"style":322},[154],[19,57662,49],{"className":57663},[75,76],[19,57665,57666,57669],{"style":387},[19,57667],{"className":57668,"style":322},[154],[19,57670,57672],{"className":57671,"style":29053},[29052],[19,57673,28986],{"className":57674},[75],[19,57676,29161],{"className":57677,"style":16031},[88,16030],[19,57679,57680,57683],{"style":504},[19,57681],{"className":57682,"style":322},[154],[19,57684],{"className":57685,"style":512},[511],[19,57687,57688,57691],{"style":515},[19,57689],{"className":57690,"style":322},[154],[19,57692,57694],{"className":57693},[75],[19,57695,3297],{"className":57696},[75,76],[19,57698,173],{"className":57699},[172],[19,57701,57703],{"className":57702},[142],[19,57704,57706],{"className":57705,"style":11497},[146],[19,57707],{},[19,57709],{"className":57710},[88,422],[19,57712],{"className":57713,"style":412},[411],[19,57715,57717,57720,57763,57823],{"className":57716},[16026],[19,57718,29161],{"className":57719,"style":16031},[81,16030],[19,57721,57723,57726],{"className":57722},[75],[19,57724,3484],{"className":57725},[75,76],[19,57727,57729],{"className":57728},[133],[19,57730,57732,57755],{"className":57731},[137,138],[19,57733,57735,57752],{"className":57734},[142],[19,57736,57738],{"className":57737,"style":3609},[146],[19,57739,57740,57743],{"style":150},[19,57741],{"className":57742,"style":155},[154],[19,57744,57746],{"className":57745},[159,160,161,162],[19,57747,57749],{"className":57748},[75,162],[19,57750,3297],{"className":57751},[75,76,162],[19,57753,173],{"className":57754},[172],[19,57756,57758],{"className":57757},[142],[19,57759,57761],{"className":57760,"style":180},[146],[19,57762],{},[19,57764,57766,57797],{"className":57765},[75],[19,57767,57769],{"className":57768},[75,29025],[19,57770,57772],{"className":57771},[137],[19,57773,57775],{"className":57774},[142],[19,57776,57778,57786],{"className":57777,"style":29035},[146],[19,57779,57780,57783],{"style":387},[19,57781],{"className":57782,"style":322},[154],[19,57784,49],{"className":57785},[75,76],[19,57787,57788,57791],{"style":387},[19,57789],{"className":57790,"style":322},[154],[19,57792,57794],{"className":57793,"style":29053},[29052],[19,57795,28986],{"className":57796},[75],[19,57798,57800],{"className":57799},[133],[19,57801,57803],{"className":57802},[137],[19,57804,57806],{"className":57805},[142],[19,57807,57809],{"className":57808,"style":4148},[146],[19,57810,57811,57814],{"style":648},[19,57812],{"className":57813,"style":155},[154],[19,57815,57817],{"className":57816},[159,160,161,162],[19,57818,57820],{"className":57819},[75,162],[19,57821,3297],{"className":57822},[75,76,162],[19,57824,29161],{"className":57825,"style":16031},[88,16030],[19,57827],{"className":57828,"style":613},[411],[19,57830,45059],{"className":57831},[335],[19,57833],{"className":57834,"style":613},[411],[19,57836,57838,57841],{"className":57837},[66],[19,57839],{"className":57840,"style":71},[70],[19,57842,57844,57847,57890,57950],{"className":57843},[16026],[19,57845,29161],{"className":57846,"style":16031},[81,16030],[19,57848,57850,57853],{"className":57849},[75],[19,57851,3484],{"className":57852},[75,76],[19,57854,57856],{"className":57855},[133],[19,57857,57859,57882],{"className":57858},[137,138],[19,57860,57862,57879],{"className":57861},[142],[19,57863,57865],{"className":57864,"style":3609},[146],[19,57866,57867,57870],{"style":150},[19,57868],{"className":57869,"style":155},[154],[19,57871,57873],{"className":57872},[159,160,161,162],[19,57874,57876],{"className":57875},[75,162],[19,57877,3297],{"className":57878},[75,76,162],[19,57880,173],{"className":57881},[172],[19,57883,57885],{"className":57884},[142],[19,57886,57888],{"className":57887,"style":180},[146],[19,57889],{},[19,57891,57893,57924],{"className":57892},[75],[19,57894,57896],{"className":57895},[75,29025],[19,57897,57899],{"className":57898},[137],[19,57900,57902],{"className":57901},[142],[19,57903,57905,57913],{"className":57904,"style":29035},[146],[19,57906,57907,57910],{"style":387},[19,57908],{"className":57909,"style":322},[154],[19,57911,49],{"className":57912},[75,76],[19,57914,57915,57918],{"style":387},[19,57916],{"className":57917,"style":322},[154],[19,57919,57921],{"className":57920,"style":29053},[29052],[19,57922,28986],{"className":57923},[75],[19,57925,57927],{"className":57926},[133],[19,57928,57930],{"className":57929},[137],[19,57931,57933],{"className":57932},[142],[19,57934,57936],{"className":57935,"style":4148},[146],[19,57937,57938,57941],{"style":648},[19,57939],{"className":57940,"style":155},[154],[19,57942,57944],{"className":57943},[159,160,161,162],[19,57945,57947],{"className":57946},[75,162],[19,57948,3297],{"className":57949},[75,76,162],[19,57951,29161],{"className":57952,"style":16031},[88,16030],[15,57954,57955,57956,57104,57994,58076,58077,58115,58116,58172,58173,56520,58211,56549,58239,58304,58305,13643,58343,51584],{},"By the comparison test, power series ",[19,57957,57959,57976],{"className":57958,"translate":23},[22],[19,57960,57962],{"className":57961},[27],[29,57963,57964],{"xmlns":31},[33,57965,57966,57974],{},[36,57967,57968,57970,57972],{},[43,57969,46],{"stretchy":45},[108,57971,3495],{},[43,57973,52],{"stretchy":45},[54,57975,51363],{"encoding":56},[19,57977,57979],{"className":57978,"ariaHidden":62},[61],[19,57980,57982,57985,57988,57991],{"className":57981},[66],[19,57983],{"className":57984,"style":71},[70],[19,57986,46],{"className":57987},[81],[19,57989,3495],{"className":57990},[75],[19,57992,52],{"className":57993},[88],[19,57995,57997,58018],{"className":57996,"translate":23},[22],[19,57998,58000],{"className":57999},[27],[29,58001,58002],{"xmlns":31},[33,58003,58004,58016],{},[36,58005,58006,58008,58010],{},[39,58007,49],{},[43,58009,267],{},[28980,58011,58012,58014],{"accent":62},[39,58013,49],{},[43,58015,28986],{},[54,58017,32971],{"encoding":56},[19,58019,58021,58039],{"className":58020,"ariaHidden":62},[61],[19,58022,58024,58027,58030,58033,58036],{"className":58023},[66],[19,58025],{"className":58026,"style":3309},[70],[19,58028,49],{"className":58029},[75,76],[19,58031],{"className":58032,"style":613},[411],[19,58034,267],{"className":58035},[335],[19,58037],{"className":58038,"style":613},[411],[19,58040,58042,58045],{"className":58041},[66],[19,58043],{"className":58044,"style":29035},[70],[19,58046,58048],{"className":58047},[75,29025],[19,58049,58051],{"className":58050},[137],[19,58052,58054],{"className":58053},[142],[19,58055,58057,58065],{"className":58056,"style":29035},[146],[19,58058,58059,58062],{"style":387},[19,58060],{"className":58061,"style":322},[154],[19,58063,49],{"className":58064},[75,76],[19,58066,58067,58070],{"style":387},[19,58068],{"className":58069,"style":322},[154],[19,58071,58073],{"className":58072,"style":29053},[29052],[19,58074,28986],{"className":58075},[75],". This contradicts the fact that the convergence interval of power series ",[19,58078,58080,58097],{"className":58079,"translate":23},[22],[19,58081,58083],{"className":58082},[27],[29,58084,58085],{"xmlns":31},[33,58086,58087,58095],{},[36,58088,58089,58091,58093],{},[43,58090,46],{"stretchy":45},[108,58092,3495],{},[43,58094,52],{"stretchy":45},[54,58096,51363],{"encoding":56},[19,58098,58100],{"className":58099,"ariaHidden":62},[61],[19,58101,58103,58106,58109,58112],{"className":58102},[66],[19,58104],{"className":58105,"style":71},[70],[19,58107,46],{"className":58108},[81],[19,58110,3495],{"className":58111},[75],[19,58113,52],{"className":58114},[88]," is ",[19,58117,58119,58142],{"className":58118,"translate":23},[22],[19,58120,58122],{"className":58121},[27],[29,58123,58124],{"xmlns":31},[33,58125,58126,58140],{},[36,58127,58128,58130,58132,58134,58136,58138],{},[43,58129,46],{"stretchy":45},[43,58131,240],{},[39,58133,10758],{},[43,58135,7830],{"separator":62},[39,58137,10758],{},[43,58139,52],{"stretchy":45},[54,58141,33497],{"encoding":56},[19,58143,58145],{"className":58144,"ariaHidden":62},[61],[19,58146,58148,58151,58154,58157,58160,58163,58166,58169],{"className":58147},[66],[19,58149],{"className":58150,"style":71},[70],[19,58152,46],{"className":58153},[81],[19,58155,240],{"className":58156},[75],[19,58158,10758],{"className":58159,"style":10844},[75,76],[19,58161,7830],{"className":58162},[8036],[19,58164],{"className":58165,"style":412},[411],[19,58167,10758],{"className":58168,"style":10844},[75,76],[19,58170,52],{"className":58171},[88],". Thus, power series ",[19,58174,58176,58193],{"className":58175,"translate":23},[22],[19,58177,58179],{"className":58178},[27],[29,58180,58181],{"xmlns":31},[33,58182,58183,58191],{},[36,58184,58185,58187,58189],{},[43,58186,46],{"stretchy":45},[108,58188,3518],{},[43,58190,52],{"stretchy":45},[54,58192,51402],{"encoding":56},[19,58194,58196],{"className":58195,"ariaHidden":62},[61],[19,58197,58199,58202,58205,58208],{"className":58198},[66],[19,58200],{"className":58201,"style":71},[70],[19,58203,46],{"className":58204},[81],[19,58206,3518],{"className":58207},[75],[19,58209,52],{"className":58210},[88],[19,58212,58214,58227],{"className":58213,"translate":23},[22],[19,58215,58217],{"className":58216},[27],[29,58218,58219],{"xmlns":31},[33,58220,58221,58225],{},[36,58222,58223],{},[39,58224,49],{},[54,58226,49],{"encoding":56},[19,58228,58230],{"className":58229,"ariaHidden":62},[61],[19,58231,58233,58236],{"className":58232},[66],[19,58234],{"className":58235,"style":3309},[70],[19,58237,49],{"className":58238},[75,76],[19,58240,58242,58265],{"className":58241,"translate":23},[22],[19,58243,58245],{"className":58244},[27],[29,58246,58247],{"xmlns":31},[33,58248,58249,58263],{},[36,58250,58251,58259,58261],{},[36,58252,58253,58255,58257],{},[43,58254,29161],{"fence":62},[39,58256,49],{},[43,58258,29161],{"fence":62},[43,58260,29448],{},[39,58262,10758],{},[54,58264,56576],{"encoding":56},[19,58266,58268,58295],{"className":58267,"ariaHidden":62},[61],[19,58269,58271,58274,58286,58289,58292],{"className":58270},[66],[19,58272],{"className":58273,"style":71},[70],[19,58275,58277,58280,58283],{"className":58276},[16026],[19,58278,29161],{"className":58279,"style":16031},[81,16030],[19,58281,49],{"className":58282},[75,76],[19,58284,29161],{"className":58285,"style":16031},[88,16030],[19,58287],{"className":58288,"style":613},[411],[19,58290,29448],{"className":58291},[335],[19,58293],{"className":58294,"style":613},[411],[19,58296,58298,58301],{"className":58297},[66],[19,58299],{"className":58300,"style":7738},[70],[19,58302,10758],{"className":58303,"style":10844},[75,76],". In summary, power series ",[19,58306,58308,58325],{"className":58307,"translate":23},[22],[19,58309,58311],{"className":58310},[27],[29,58312,58313],{"xmlns":31},[33,58314,58315,58323],{},[36,58316,58317,58319,58321],{},[43,58318,46],{"stretchy":45},[108,58320,3495],{},[43,58322,52],{"stretchy":45},[54,58324,51363],{"encoding":56},[19,58326,58328],{"className":58327,"ariaHidden":62},[61],[19,58329,58331,58334,58337,58340],{"className":58330},[66],[19,58332],{"className":58333,"style":71},[70],[19,58335,46],{"className":58336},[81],[19,58338,3495],{"className":58339},[75],[19,58341,52],{"className":58342},[88],[19,58344,58346,58363],{"className":58345,"translate":23},[22],[19,58347,58349],{"className":58348},[27],[29,58350,58351],{"xmlns":31},[33,58352,58353,58361],{},[36,58354,58355,58357,58359],{},[43,58356,46],{"stretchy":45},[108,58358,3518],{},[43,58360,52],{"stretchy":45},[54,58362,51402],{"encoding":56},[19,58364,58366],{"className":58365,"ariaHidden":62},[61],[19,58367,58369,58372,58375,58378],{"className":58368},[66],[19,58370],{"className":58371,"style":71},[70],[19,58373,46],{"className":58374},[81],[19,58376,3518],{"className":58377},[75],[19,58379,52],{"className":58380},[88],[15,58382,58383],{},"Regarding term-by-term differentiation and integration of power series, we have the following theorem:",[15,58385,58386,58388,58389,58115,58427,58484,58485,58528,58529,58557,58558,58614],{},[7600,58387,9131],{}," Suppose the convergence interval of series ",[19,58390,58392,58409],{"className":58391,"translate":23},[22],[19,58393,58395],{"className":58394},[27],[29,58396,58397],{"xmlns":31},[33,58398,58399,58407],{},[36,58400,58401,58403,58405],{},[43,58402,46],{"stretchy":45},[108,58404,3495],{},[43,58406,52],{"stretchy":45},[54,58408,51363],{"encoding":56},[19,58410,58412],{"className":58411,"ariaHidden":62},[61],[19,58413,58415,58418,58421,58424],{"className":58414},[66],[19,58416],{"className":58417,"style":71},[70],[19,58419,46],{"className":58420},[81],[19,58422,3495],{"className":58423},[75],[19,58425,52],{"className":58426},[88],[19,58428,58430,58454],{"className":58429,"translate":23},[22],[19,58431,58433],{"className":58432},[27],[29,58434,58435],{"xmlns":31},[33,58436,58437,58451],{},[36,58438,58439,58441,58443,58445,58447,58449],{},[43,58440,46],{"stretchy":45},[43,58442,240],{},[39,58444,10758],{},[43,58446,7830],{"separator":62},[39,58448,10758],{},[43,58450,52],{"stretchy":45},[54,58452,58453],{"encoding":56},"(-R, R)",[19,58455,58457],{"className":58456,"ariaHidden":62},[61],[19,58458,58460,58463,58466,58469,58472,58475,58478,58481],{"className":58459},[66],[19,58461],{"className":58462,"style":71},[70],[19,58464,46],{"className":58465},[81],[19,58467,240],{"className":58468},[75],[19,58470,10758],{"className":58471,"style":10844},[75,76],[19,58473,7830],{"className":58474},[8036],[19,58476],{"className":58477,"style":412},[411],[19,58479,10758],{"className":58480,"style":10844},[75,76],[19,58482,52],{"className":58483},[88],", and its sum function on this interval is denoted by ",[19,58486,58488,58507],{"className":58487,"translate":23},[22],[19,58489,58491],{"className":58490},[27],[29,58492,58493],{"xmlns":31},[33,58494,58495,58505],{},[36,58496,58497,58499,58501,58503],{},[39,58498,41],{},[43,58500,46],{"stretchy":45},[39,58502,49],{},[43,58504,52],{"stretchy":45},[54,58506,57],{"encoding":56},[19,58508,58510],{"className":58509,"ariaHidden":62},[61],[19,58511,58513,58516,58519,58522,58525],{"className":58512},[66],[19,58514],{"className":58515,"style":71},[70],[19,58517,41],{"className":58518,"style":77},[75,76],[19,58520,46],{"className":58521},[81],[19,58523,49],{"className":58524},[75,76],[19,58526,52],{"className":58527},[88],". If ",[19,58530,58532,58545],{"className":58531,"translate":23},[22],[19,58533,58535],{"className":58534},[27],[29,58536,58537],{"xmlns":31},[33,58538,58539,58543],{},[36,58540,58541],{},[39,58542,49],{},[54,58544,49],{"encoding":56},[19,58546,58548],{"className":58547,"ariaHidden":62},[61],[19,58549,58551,58554],{"className":58550},[66],[19,58552],{"className":58553,"style":3309},[70],[19,58555,49],{"className":58556},[75,76]," is any point in ",[19,58559,58561,58584],{"className":58560,"translate":23},[22],[19,58562,58564],{"className":58563},[27],[29,58565,58566],{"xmlns":31},[33,58567,58568,58582],{},[36,58569,58570,58572,58574,58576,58578,58580],{},[43,58571,46],{"stretchy":45},[43,58573,240],{},[39,58575,10758],{},[43,58577,7830],{"separator":62},[39,58579,10758],{},[43,58581,52],{"stretchy":45},[54,58583,58453],{"encoding":56},[19,58585,58587],{"className":58586,"ariaHidden":62},[61],[19,58588,58590,58593,58596,58599,58602,58605,58608,58611],{"className":58589},[66],[19,58591],{"className":58592,"style":71},[70],[19,58594,46],{"className":58595},[81],[19,58597,240],{"className":58598},[75],[19,58600,10758],{"className":58601,"style":10844},[75,76],[19,58603,7830],{"className":58604},[8036],[19,58606],{"className":58607,"style":412},[411],[19,58609,10758],{"className":58610,"style":10844},[75,76],[19,58612,52],{"className":58613},[88],", then:",[34217,58616,58617,59019],{},[34220,58618,58619,34262,58657,89,58700,58728,58729],{},[19,58620,58622,58639],{"className":58621,"translate":23},[22],[19,58623,58625],{"className":58624},[27],[29,58626,58627],{"xmlns":31},[33,58628,58629,58637],{},[36,58630,58631,58633,58635],{},[43,58632,46],{"stretchy":45},[39,58634,34238],{},[43,58636,52],{"stretchy":45},[54,58638,34243],{"encoding":56},[19,58640,58642],{"className":58641,"ariaHidden":62},[61],[19,58643,58645,58648,58651,58654],{"className":58644},[66],[19,58646],{"className":58647,"style":71},[70],[19,58649,46],{"className":58650},[81],[19,58652,34238],{"className":58653},[75,76],[19,58655,52],{"className":58656},[88],[19,58658,58660,58679],{"className":58659,"translate":23},[22],[19,58661,58663],{"className":58662},[27],[29,58664,58665],{"xmlns":31},[33,58666,58667,58677],{},[36,58668,58669,58671,58673,58675],{},[39,58670,41],{},[43,58672,46],{"stretchy":45},[39,58674,49],{},[43,58676,52],{"stretchy":45},[54,58678,57],{"encoding":56},[19,58680,58682],{"className":58681,"ariaHidden":62},[61],[19,58683,58685,58688,58691,58694,58697],{"className":58684},[66],[19,58686],{"className":58687,"style":71},[70],[19,58689,41],{"className":58690,"style":77},[75,76],[19,58692,46],{"className":58693},[81],[19,58695,49],{"className":58696},[75,76],[19,58698,52],{"className":58699},[88],[19,58701,58703,58716],{"className":58702,"translate":23},[22],[19,58704,58706],{"className":58705},[27],[29,58707,58708],{"xmlns":31},[33,58709,58710,58714],{},[36,58711,58712],{},[39,58713,49],{},[54,58715,49],{"encoding":56},[19,58717,58719],{"className":58718,"ariaHidden":62},[61],[19,58720,58722,58725],{"className":58721},[66],[19,58723],{"className":58724,"style":3309},[70],[19,58726,49],{"className":58727},[75,76],", and\n",[19,58730,58732],{"className":58731,"translate":23},[187],[19,58733,58735,58797],{"className":58734,"translate":23},[22],[19,58736,58738],{"className":58737},[27],[29,58739,58740],{"xmlns":31,"display":196},[33,58741,58742,58794],{},[36,58743,58744,58750,58752,58754,58756,58758,58772],{},[269,58745,58746,58748],{},[39,58747,41],{},[43,58749,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,58751,46],{"stretchy":45},[39,58753,49],{},[43,58755,52],{"stretchy":45},[43,58757,267],{},[26840,58759,58760,58762,58770],{},[43,58761,26844],{},[36,58763,58764,58766,58768],{},[39,58765,3297],{},[43,58767,267],{},[108,58769,3495],{},[39,58771,26855],{"mathvariant":275},[36,58773,58774,58776,58782],{},[39,58775,3297],{},[103,58777,58778,58780],{},[39,58779,3484],{},[39,58781,3297],{},[269,58783,58784,58786],{},[39,58785,49],{},[36,58787,58788,58790,58792],{},[39,58789,3297],{},[43,58791,240],{},[108,58793,3495],{},[54,58795,58796],{"encoding":56},"f'(x)=\\sum_{n=1}^{\\infty}{na_{n}x^{n-1}}",[19,58798,58800,58856],{"className":58799,"ariaHidden":62},[61],[19,58801,58803,58806,58838,58841,58844,58847,58850,58853],{"className":58802},[66],[19,58804],{"className":58805,"style":626},[70],[19,58807,58809,58812],{"className":58808},[75],[19,58810,41],{"className":58811,"style":77},[75,76],[19,58813,58815],{"className":58814},[133],[19,58816,58818],{"className":58817},[137],[19,58819,58821],{"className":58820},[142],[19,58822,58824],{"className":58823,"style":645},[146],[19,58825,58826,58829],{"style":648},[19,58827],{"className":58828,"style":155},[154],[19,58830,58832],{"className":58831},[159,160,161,162],[19,58833,58835],{"className":58834},[75,162],[19,58836,277],{"className":58837},[75,162],[19,58839,46],{"className":58840},[81],[19,58842,49],{"className":58843},[75,76],[19,58845,52],{"className":58846},[88],[19,58848],{"className":58849,"style":613},[411],[19,58851,267],{"className":58852},[335],[19,58854],{"className":58855,"style":613},[411],[19,58857,58859,58862,58929,58932],{"className":58858},[66],[19,58860],{"className":58861,"style":26984},[70],[19,58863,58865],{"className":58864},[304,305],[19,58866,58868,58921],{"className":58867},[137,138],[19,58869,58871,58918],{"className":58870},[142],[19,58872,58874,58894,58904],{"className":58873,"style":26997},[146],[19,58875,58876,58879],{"style":27000},[19,58877],{"className":58878,"style":27004},[154],[19,58880,58882],{"className":58881},[159,160,161,162],[19,58883,58885,58888,58891],{"className":58884},[75,162],[19,58886,3297],{"className":58887},[75,76,162],[19,58889,267],{"className":58890},[335,162],[19,58892,3495],{"className":58893},[75,162],[19,58895,58896,58899],{"style":27022},[19,58897],{"className":58898,"style":27004},[154],[19,58900,58901],{},[19,58902,26844],{"className":58903},[304,27031,27032],[19,58905,58906,58909],{"style":27035},[19,58907],{"className":58908,"style":27004},[154],[19,58910,58912],{"className":58911},[159,160,161,162],[19,58913,58915],{"className":58914},[75,162],[19,58916,26855],{"className":58917},[75,162],[19,58919,173],{"className":58920},[172],[19,58922,58924],{"className":58923},[142],[19,58925,58927],{"className":58926,"style":27057},[146],[19,58928],{},[19,58930],{"className":58931,"style":412},[411],[19,58933,58935,58938,58981],{"className":58934},[75],[19,58936,3297],{"className":58937},[75,76],[19,58939,58941,58944],{"className":58940},[75],[19,58942,3484],{"className":58943},[75,76],[19,58945,58947],{"className":58946},[133],[19,58948,58950,58973],{"className":58949},[137,138],[19,58951,58953,58970],{"className":58952},[142],[19,58954,58956],{"className":58955,"style":3609},[146],[19,58957,58958,58961],{"style":150},[19,58959],{"className":58960,"style":155},[154],[19,58962,58964],{"className":58963},[159,160,161,162],[19,58965,58967],{"className":58966},[75,162],[19,58968,3297],{"className":58969},[75,76,162],[19,58971,173],{"className":58972},[172],[19,58974,58976],{"className":58975},[142],[19,58977,58979],{"className":58978,"style":180},[146],[19,58980],{},[19,58982,58984,58987],{"className":58983},[75],[19,58985,49],{"className":58986},[75,76],[19,58988,58990],{"className":58989},[133],[19,58991,58993],{"className":58992},[137],[19,58994,58996],{"className":58995},[142],[19,58997,58999],{"className":58998,"style":3970},[146],[19,59000,59001,59004],{"style":648},[19,59002],{"className":59003,"style":155},[154],[19,59005,59007],{"className":59006},[159,160,161,162],[19,59008,59010,59013,59016],{"className":59009},[75,162],[19,59011,3297],{"className":59012},[75,76,162],[19,59014,240],{"className":59015},[455,162],[19,59017,3495],{"className":59018},[75,162],[34220,59020,59021,34262,59061,59104,59105,13643,59133,59161,59162,58728,59218],{},[19,59022,59024,59043],{"className":59023,"translate":23},[22],[19,59025,59027],{"className":59026},[27],[29,59028,59029],{"xmlns":31},[33,59030,59031,59041],{},[36,59032,59033,59035,59037,59039],{},[43,59034,46],{"stretchy":45},[39,59036,34238],{},[39,59038,34238],{},[43,59040,52],{"stretchy":45},[54,59042,34491],{"encoding":56},[19,59044,59046],{"className":59045,"ariaHidden":62},[61],[19,59047,59049,59052,59055,59058],{"className":59048},[66],[19,59050],{"className":59051,"style":71},[70],[19,59053,46],{"className":59054},[81],[19,59056,34507],{"className":59057},[75,76],[19,59059,52],{"className":59060},[88],[19,59062,59064,59083],{"className":59063,"translate":23},[22],[19,59065,59067],{"className":59066},[27],[29,59068,59069],{"xmlns":31},[33,59070,59071,59081],{},[36,59072,59073,59075,59077,59079],{},[39,59074,41],{},[43,59076,46],{"stretchy":45},[39,59078,49],{},[43,59080,52],{"stretchy":45},[54,59082,57],{"encoding":56},[19,59084,59086],{"className":59085,"ariaHidden":62},[61],[19,59087,59089,59092,59095,59098,59101],{"className":59088},[66],[19,59090],{"className":59091,"style":71},[70],[19,59093,41],{"className":59094,"style":77},[75,76],[19,59096,46],{"className":59097},[81],[19,59099,49],{"className":59100},[75,76],[19,59102,52],{"className":59103},[88]," is integrable over the interval between ",[19,59106,59108,59121],{"className":59107,"translate":23},[22],[19,59109,59111],{"className":59110},[27],[29,59112,59113],{"xmlns":31},[33,59114,59115,59119],{},[36,59116,59117],{},[108,59118,110],{},[54,59120,110],{"encoding":56},[19,59122,59124],{"className":59123,"ariaHidden":62},[61],[19,59125,59127,59130],{"className":59126},[66],[19,59128],{"className":59129,"style":1214},[70],[19,59131,110],{"className":59132},[75],[19,59134,59136,59149],{"className":59135,"translate":23},[22],[19,59137,59139],{"className":59138},[27],[29,59140,59141],{"xmlns":31},[33,59142,59143,59147],{},[36,59144,59145],{},[39,59146,49],{},[54,59148,49],{"encoding":56},[19,59150,59152],{"className":59151,"ariaHidden":62},[61],[19,59153,59155,59158],{"className":59154},[66],[19,59156],{"className":59157,"style":3309},[70],[19,59159,49],{"className":59160},[75,76]," on ",[19,59163,59165,59188],{"className":59164,"translate":23},[22],[19,59166,59168],{"className":59167},[27],[29,59169,59170],{"xmlns":31},[33,59171,59172,59186],{},[36,59173,59174,59176,59178,59180,59182,59184],{},[43,59175,46],{"stretchy":45},[43,59177,240],{},[39,59179,10758],{},[43,59181,7830],{"separator":62},[39,59183,10758],{},[43,59185,52],{"stretchy":45},[54,59187,58453],{"encoding":56},[19,59189,59191],{"className":59190,"ariaHidden":62},[61],[19,59192,59194,59197,59200,59203,59206,59209,59212,59215],{"className":59193},[66],[19,59195],{"className":59196,"style":71},[70],[19,59198,46],{"className":59199},[81],[19,59201,240],{"className":59202},[75],[19,59204,10758],{"className":59205,"style":10844},[75,76],[19,59207,7830],{"className":59208},[8036],[19,59210],{"className":59211,"style":412},[411],[19,59213,10758],{"className":59214,"style":10844},[75,76],[19,59216,52],{"className":59217},[88],[19,59219,59221],{"className":59220,"translate":23},[187],[19,59222,59224,59303],{"className":59223,"translate":23},[22],[19,59225,59227],{"className":59226},[27],[29,59228,59229],{"xmlns":31,"display":196},[33,59230,59231,59300],{},[36,59232,59233,59241,59243,59245,59248,59250,59252,59254,59256,59270],{},[4285,59234,59235,59237,59239],{},[43,59236,48238],{},[108,59238,110],{},[39,59240,49],{},[39,59242,41],{},[43,59244,46],{"stretchy":45},[39,59246,59247],{},"t",[43,59249,52],{"stretchy":45},[39,59251,48257],{},[39,59253,59247],{},[43,59255,267],{},[26840,59257,59258,59260,59268],{},[43,59259,26844],{},[36,59261,59262,59264,59266],{},[39,59263,3297],{},[43,59265,267],{},[108,59267,110],{},[39,59269,26855],{"mathvariant":275},[36,59271,59272,59288],{},[226,59273,59274,59280],{},[103,59275,59276,59278],{},[39,59277,3484],{},[39,59279,3297],{},[36,59281,59282,59284,59286],{},[39,59283,3297],{},[43,59285,2351],{},[108,59287,3495],{},[269,59289,59290,59292],{},[39,59291,49],{},[36,59293,59294,59296,59298],{},[39,59295,3297],{},[43,59297,2351],{},[108,59299,3495],{},[54,59301,59302],{"encoding":56},"\\int_{0}^{x}f(t)dt=\\sum_{n=0}^{\\infty}{\\frac{a_{n}}{n+1}x^{n+1}}",[19,59304,59306,59401],{"className":59305,"ariaHidden":62},[61],[19,59307,59309,59313,59371,59374,59377,59380,59383,59386,59389,59392,59395,59398],{"className":59308},[66],[19,59310],{"className":59311,"style":59312},[70],"height:2.3262em;vertical-align:-0.9119em;",[19,59314,59316,59319],{"className":59315},[304],[19,59317,48238],{"className":59318,"style":48393},[304,27031,27032],[19,59320,59322],{"className":59321},[133],[19,59323,59325,59363],{"className":59324},[137,138],[19,59326,59328,59360],{"className":59327},[142],[19,59329,59332,59346],{"className":59330,"style":59331},[146],"height:1.4143em;",[19,59333,59334,59337],{"style":48409},[19,59335],{"className":59336,"style":155},[154],[19,59338,59340],{"className":59339},[159,160,161,162],[19,59341,59343],{"className":59342},[75,162],[19,59344,110],{"className":59345},[75,162],[19,59347,59348,59351],{"style":48424},[19,59349],{"className":59350,"style":155},[154],[19,59352,59354],{"className":59353},[159,160,161,162],[19,59355,59357],{"className":59356},[75,162],[19,59358,49],{"className":59359},[75,76,162],[19,59361,173],{"className":59362},[172],[19,59364,59366],{"className":59365},[142],[19,59367,59369],{"className":59368,"style":48446},[146],[19,59370],{},[19,59372],{"className":59373,"style":412},[411],[19,59375,41],{"className":59376,"style":77},[75,76],[19,59378,46],{"className":59379},[81],[19,59381,59247],{"className":59382},[75,76],[19,59384,52],{"className":59385},[88],[19,59387,48257],{"className":59388},[75,76],[19,59390,59247],{"className":59391},[75,76],[19,59393],{"className":59394,"style":613},[411],[19,59396,267],{"className":59397},[335],[19,59399],{"className":59400,"style":613},[411],[19,59402,59404,59407,59474,59477],{"className":59403},[66],[19,59405],{"className":59406,"style":26984},[70],[19,59408,59410],{"className":59409},[304,305],[19,59411,59413,59466],{"className":59412},[137,138],[19,59414,59416,59463],{"className":59415},[142],[19,59417,59419,59439,59449],{"className":59418,"style":26997},[146],[19,59420,59421,59424],{"style":27000},[19,59422],{"className":59423,"style":27004},[154],[19,59425,59427],{"className":59426},[159,160,161,162],[19,59428,59430,59433,59436],{"className":59429},[75,162],[19,59431,3297],{"className":59432},[75,76,162],[19,59434,267],{"className":59435},[335,162],[19,59437,110],{"className":59438},[75,162],[19,59440,59441,59444],{"style":27022},[19,59442],{"className":59443,"style":27004},[154],[19,59445,59446],{},[19,59447,26844],{"className":59448},[304,27031,27032],[19,59450,59451,59454],{"style":27035},[19,59452],{"className":59453,"style":27004},[154],[19,59455,59457],{"className":59456},[159,160,161,162],[19,59458,59460],{"className":59459},[75,162],[19,59461,26855],{"className":59462},[75,162],[19,59464,173],{"className":59465},[172],[19,59467,59469],{"className":59468},[142],[19,59470,59472],{"className":59471,"style":27057},[146],[19,5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Since series ",[19,59638,59640,59657],{"className":59639,"translate":23},[22],[19,59641,59643],{"className":59642},[27],[29,59644,59645],{"xmlns":31},[33,59646,59647,59655],{},[36,59648,59649,59651,59653],{},[43,59650,46],{"stretchy":45},[108,59652,3495],{},[43,59654,52],{"stretchy":45},[54,59656,51363],{"encoding":56},[19,59658,59660],{"className":59659,"ariaHidden":62},[61],[19,59661,59663,59666,59669,59672],{"className":59662},[66],[19,59664],{"className":59665,"style":71},[70],[19,59667,46],{"className":59668},[81],[19,59670,3495],{"className":59671},[75],[19,59673,52],{"className":59674},[88],[19,59676,59678,59695],{"className":59677,"translate":23},[22],[19,59679,59681],{"className":59680},[27],[29,59682,59683],{"xmlns":31},[33,59684,59685,59693],{},[36,59686,59687,59689,59691],{},[43,59688,46],{"stretchy":45},[108,59690,3518],{},[43,59692,52],{"stretchy":45},[54,59694,51402],{"encoding":56},[19,59696,59698],{"className":59697,"ariaHidden":62},[61],[19,59699,59701,59704,59707,59710],{"className":59700},[66],[19,59702],{"className":59703,"style":71},[70],[19,59705,46],{"className":59706},[81],[19,59708,3518],{"className":59709},[75],[19,59711,52],{"className":59712},[88],[19,59714,59716,59733],{"className":59715,"translate":23},[22],[19,59717,59719],{"className":59718},[27],[29,59720,59721],{"xmlns":31},[33,59722,59723,59731],{},[36,59724,59725,59727,59729],{},[43,59726,46],{"stretchy":45},[108,59728,39228],{},[43,59730,52],{"stretchy":45},[54,59732,51441],{"encoding":56},[19,59734,59736],{"className":59735,"ariaHidden":62},[61],[19,59737,59739,59742,59745,59748],{"className":59738},[66],[19,59740],{"className":59741,"style":71},[70],[19,59743,46],{"className":59744},[81],[19,59746,39228],{"className":59747},[75],[19,59749,52],{"className":59750},[88]," have the same convergence radius ",[19,59753,59755,59768],{"className":59754,"translate":23},[22],[19,59756,59758],{"className":59757},[27],[29,59759,59760],{"xmlns":31},[33,59761,59762,59766],{},[36,59763,59764],{},[39,59765,10758],{},[54,59767,10758],{"encoding":56},[19,59769,59771],{"className":59770,"ariaHidden":62},[61],[19,59772,59774,59777],{"className":59773},[66],[19,59775],{"className":59776,"style":7738},[70],[19,59778,10758],{"className":59779,"style":10844},[75,76],", and each term of series ",[19,59782,59784,59801],{"className":59783,"translate":23},[22],[19,59785,59787],{"className":59786},[27],[29,59788,59789],{"xmlns":31},[33,59790,59791,59799],{},[36,59792,59793,59795,59797],{},[43,59794,46],{"stretchy":45},[108,59796,3495],{},[43,59798,52],{"stretchy":45},[54,59800,51363],{"encoding":56},[19,59802,59804],{"className":59803,"ariaHidden":62},[61],[19,59805,59807,59810,59813,59816],{"className":59806},[66],[19,59808],{"className":59809,"style":71},[70],[19,59811,46],{"className":59812},[81],[19,59814,3495],{"className":59815},[75],[19,59817,52],{"className":59818},[88]," has a continuous derivative, and all three power series converge uniformly on the closed interval ",[19,59821,59823,59846],{"className":59822,"translate":23},[22],[19,59824,59826],{"className":59825},[27],[29,59827,59828],{"xmlns":31},[33,59829,59830,59844],{},[36,59831,59832,59834,59836,59838,59840,59842],{},[43,59833,46],{"stretchy":45},[43,59835,240],{},[39,59837,10758],{},[43,59839,7830],{"separator":62},[39,59841,10758],{},[43,59843,52],{"stretchy":45},[54,59845,58453],{"encoding":56},[19,59847,59849],{"className":59848,"ariaHidden":62},[61],[19,59850,59852,59855,59858,59861,59864,59867,59870,59873],{"className":59851},[66],[19,59853],{"className":59854,"style":71},[70],[19,59856,46],{"className":59857},[81],[19,59859,240],{"className":59860},[75],[19,59862,10758],{"className":59863,"style":10844},[75,76],[19,59865,7830],{"className":59866},[8036],[19,59868],{"className":59869,"style":412},[411],[19,59871,10758],{"className":59872,"style":10844},[75,76],[19,59874,52],{"className":59875},[88],", they satisfy the theorem of term-by-term differentiation and integration of function series. Therefore, the above theorem holds.",[15,59878,59879],{},"From the above theorem, we can derive the following corollaries:",[15,59881,59882,42193,59885,59928,59929,59967,59968,60024,60025,60068,60069,60125],{},[7600,59883,59884],{},"Corollary 1:",[19,59886,59888,59907],{"className":59887,"translate":23},[22],[19,59889,59891],{"className":59890},[27],[29,59892,59893],{"xmlns":31},[33,59894,59895,59905],{},[36,59896,59897,59899,59901,59903],{},[39,59898,41],{},[43,59900,46],{"stretchy":45},[39,59902,49],{},[43,59904,52],{"stretchy":45},[54,59906,57],{"encoding":56},[19,59908,59910],{"className":59909,"ariaHidden":62},[61],[19,59911,59913,59916,59919,59922,59925],{"className":59912},[66],[19,59914],{"className":59915,"style":71},[70],[19,59917,41],{"className":59918,"style":77},[75,76],[19,59920,46],{"className":59921},[81],[19,59923,49],{"className":59924},[75,76],[19,59926,52],{"className":59927},[88]," be the sum function of series ",[19,59930,59932,59949],{"className":59931,"translate":23},[22],[19,59933,59935],{"className":59934},[27],[29,59936,59937],{"xmlns":31},[33,59938,59939,59947],{},[36,59940,59941,59943,59945],{},[43,59942,46],{"stretchy":45},[108,59944,3495],{},[43,59946,52],{"stretchy":45},[54,59948,51363],{"encoding":56},[19,59950,59952],{"className":59951,"ariaHidden":62},[61],[19,59953,59955,59958,59961,59964],{"className":59954},[66],[19,59956],{"className":59957,"style":71},[70],[19,59959,46],{"className":59960},[81],[19,59962,3495],{"className":59963},[75],[19,59965,52],{"className":59966},[88]," on the convergence interval ",[19,59969,59971,59994],{"className":59970,"translate":23},[22],[19,59972,59974],{"className":59973},[27],[29,59975,59976],{"xmlns":31},[33,59977,59978,59992],{},[36,59979,59980,59982,59984,59986,59988,59990],{},[43,59981,46],{"stretchy":45},[43,59983,240],{},[39,59985,10758],{},[43,59987,7830],{"separator":62},[39,59989,10758],{},[43,59991,52],{"stretchy":45},[54,59993,58453],{"encoding":56},[19,59995,59997],{"className":59996,"ariaHidden":62},[61],[19,59998,60000,60003,60006,60009,60012,60015,60018,60021],{"className":59999},[66],[19,60001],{"className":60002,"style":71},[70],[19,60004,46],{"className":60005},[81],[19,60007,240],{"className":60008},[75],[19,60010,10758],{"className":60011,"style":10844},[75,76],[19,60013,7830],{"className":60014},[8036],[19,60016],{"className":60017,"style":412},[411],[19,60019,10758],{"className":60020,"style":10844},[75,76],[19,60022,52],{"className":60023},[88],". Then, ",[19,60026,60028,60047],{"className":60027,"translate":23},[22],[19,60029,60031],{"className":60030},[27],[29,60032,60033],{"xmlns":31},[33,60034,60035,60045],{},[36,60036,60037,60039,60041,60043],{},[39,60038,41],{},[43,60040,46],{"stretchy":45},[39,60042,49],{},[43,60044,52],{"stretchy":45},[54,60046,57],{"encoding":56},[19,60048,60050],{"className":60049,"ariaHidden":62},[61],[19,60051,60053,60056,60059,60062,60065],{"className":60052},[66],[19,60054],{"className":60055,"style":71},[70],[19,60057,41],{"className":60058,"style":77},[75,76],[19,60060,46],{"className":60061},[81],[19,60063,49],{"className":60064},[75,76],[19,60066,52],{"className":60067},[88]," has derivatives of any order on ",[19,60070,60072,60095],{"className":60071,"translate":23},[22],[19,60073,60075],{"className":60074},[27],[29,60076,60077],{"xmlns":31},[33,60078,60079,60093],{},[36,60080,60081,60083,60085,60087,60089,60091],{},[43,60082,46],{"stretchy":45},[43,60084,240],{},[39,60086,10758],{},[43,60088,7830],{"separator":62},[39,60090,10758],{},[43,60092,52],{"stretchy":45},[54,60094,58453],{"encoding":56},[19,60096,60098],{"className":60097,"ariaHidden":62},[61],[19,60099,60101,60104,60107,60110,60113,60116,60119,60122],{"className":60100},[66],[19,60102],{"className":60103,"style":71},[70],[19,60105,46],{"className":60106},[81],[19,60108,240],{"className":60109},[75],[19,60111,10758],{"className":60112,"style":10844},[75,76],[19,60114,7830],{"className":60115},[8036],[19,60117],{"className":60118,"style":412},[411],[19,60120,10758],{"className":60121,"style":10844},[75,76],[19,60123,52],{"className":60124},[88],", and can be differentiated term by term any number of times, 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\\\\ a_{n}=\\frac{f^{(n)}(0)}{n!} \\quad (n=1,2,...)",[19,61833,61835,61893,61911,61914,61972,62105],{"className":61834,"ariaHidden":62},[61],[19,61836,61838,61841,61884,61887,61890],{"className":61837},[66],[19,61839],{"className":61840,"style":123},[70],[19,61842,61844,61847],{"className":61843},[75],[19,61845,3484],{"className":61846},[75,76],[19,61848,61850],{"className":61849},[133],[19,61851,61853,61876],{"className":61852},[137,138],[19,61854,61856,61873],{"className":61855},[142],[19,61857,61859],{"className":61858,"style":147},[146],[19,61860,61861,61864],{"style":150},[19,61862],{"className":61863,"style":155},[154],[19,61865,61867],{"className":61866},[159,160,161,162],[19,61868,61870],{"className":61869},[75,162],[19,61871,110],{"className":61872},[75,162],[19,61874,173],{"className":61875},[172],[19,61877,61879],{"className":61878},[142],[19,61880,61882],{"className":61881,"style":180},[146],[19,61883],{},[19,61885],{"className":61886,"style":613},[411],[19,61888,267],{"className":61889},[335],[19,61891],{"className":61892,"style":613},[411],[19,61894,61896,61899,61902,61905,61908],{"className":61895},[66],[19,61897],{"className":61898,"style":71},[70],[19,61900,41],{"className":61901,"style":77},[75,76],[19,61903,46],{"className":61904},[81],[19,61906,110],{"className":61907},[75],[19,61909,52],{"className":61910},[88],[19,61912],{"className":61913},[411,4283],[19,61915,61917,61920,61963,61966,61969],{"className":61916},[66],[19,61918],{"className":61919,"style":123},[70],[19,61921,61923,61926],{"className":61922},[75],[19,61924,3484],{"className":61925},[75,76],[19,61927,61929],{"className":61928},[133],[19,61930,61932,61955],{"className":61931},[137,138],[19,61933,61935,61952],{"className":61934},[142],[19,61936,61938],{"className":61937,"style":3609},[146],[19,61939,61940,61943],{"style":150},[19,61941],{"className":61942,"style":155},[154],[19,61944,61946],{"className":61945},[159,160,161,162],[19,61947,61949],{"className":61948},[75,162],[19,61950,3297],{"className":61951},[75,76,162],[19,61953,173],{"className":61954},[172],[19,61956,61958],{"className":61957},[142],[19,61959,61961],{"className":61960,"style":180},[146],[19,61962],{},[19,61964],{"className":61965,"style":613},[411],[19,61967,267],{"className":61968},[335],[19,61970],{"className":61971,"style":613},[411],[19,61973,61975,61978,62087,62090,62093,62096,62099,62102],{"className":61974},[66],[19,61976],{"className":61977,"style":7348},[70],[19,61979,61981,61984,62084],{"className":61980},[75],[19,61982],{"className":61983},[81,422],[19,61985,61987],{"className":61986},[226],[19,61988,61990,62076],{"className":61989},[137,138],[19,61991,61993,62073],{"className":61992},[142],[19,61994,61996,62010,62018],{"className":61995,"style":7367},[146],[19,61997,61998,62001],{"style":438},[19,61999],{"className":62000,"style":322},[154],[19,62002,62004,62007],{"className":62003},[75],[19,62005,3297],{"className":62006},[75,76],[19,62008,4341],{"className":62009},[88],[19,62011,62012,62015],{"style":504},[19,62013],{"className":62014,"style":322},[154],[19,62016],{"className":62017,"style":512},[511],[19,62019,62020,62023],{"style":515},[19,62021],{"className":62022,"style":322},[154],[19,62024,62026,62064,62067,62070],{"className":62025},[75],[19,62027,62029,62032],{"className":62028},[75],[19,62030,41],{"className":62031,"style":77},[75,76],[19,62033,62035],{"className":62034},[133],[19,62036,62038],{"className":62037},[137],[19,62039,62041],{"className":62040},[142],[19,62042,62044],{"className":62043,"style":7416},[146],[19,62045,62046,62049],{"style":1643},[19,62047],{"className":62048,"style":155},[154],[19,62050,62052],{"className":62051},[159,160,161,162],[19,62053,62055,62058,62061],{"className":62054},[75,162],[19,62056,46],{"className":62057},[81,162],[19,62059,3297],{"className":62060},[75,76,162],[19,62062,52],{"className":62063},[88,162],[19,62065,46],{"className":62066},[81],[19,62068,110],{"className":62069},[75],[19,62071,52],{"className":62072},[88],[19,62074,173],{"className":62075},[172],[19,62077,62079],{"className":62078},[142],[19,62080,62082],{"className":62081,"style":5671},[146],[19,62083],{},[19,62085],{"className":62086},[88,422],[19,62088],{"className":62089,"style":30184},[411],[19,62091,46],{"className":62092},[81],[19,62094,3297],{"className":62095},[75,76],[19,62097],{"className":62098,"style":613},[411],[19,62100,267],{"className":62101},[335],[19,62103],{"className":62104,"style":613},[411],[19,62106,62108,62111,62114,62117,62120,62123,62126,62129,62132],{"className":62107},[66],[19,62109],{"className":62110,"style":71},[70],[19,62112,3495],{"className":62113},[75],[19,62115,7830],{"className":62116},[8036],[19,62118],{"className":62119,"style":412},[411],[19,62121,3518],{"className":62122},[75],[19,62124,7830],{"className":62125},[8036],[19,62127],{"className":62128,"style":412},[411],[19,62130,4007],{"className":62131},[75],[19,62133,52],{"className":62134},[88],[15,62136,62137,43313,62140,13643,62335,62531,62532,62582],{},[7600,62138,62139],{},"Equality of Power Series:",[19,62141,62143,62182],{"className":62142,"translate":23},[22],[19,62144,62146],{"className":62145},[27],[29,62147,62148],{"xmlns":31},[33,62149,62150,62180],{},[36,62151,62152,62166],{},[4285,62153,62154,62156,62164],{},[43,62155,26844],{},[36,62157,62158,62160,62162],{},[39,62159,3297],{},[43,62161,267],{},[108,62163,110],{},[39,62165,26855],{"mathvariant":275},[36,62167,62168,62174],{},[103,62169,62170,62172],{},[39,62171,3484],{},[39,62173,3297],{},[269,62175,62176,62178],{},[39,62177,49],{},[39,62179,3297],{},[54,62181,28591],{"encoding":56},[19,62183,62185],{"className":62184,"ariaHidden":62},[61],[19,62186,62188,62191,62254,62257],{"className":62187},[66],[19,62189],{"className":62190,"style":31929},[70],[19,62192,62194,62197],{"className":62193},[304],[19,62195,26844],{"className":62196,"style":31937},[304,27031,31936],[19,62198,62200],{"className":62199},[133],[19,62201,62203,62246],{"className":62202},[137,138],[19,62204,62206,62243],{"className":62205},[142],[19,62207,62209,62229],{"className":62208,"style":31950},[146],[19,62210,62211,62214],{"style":31953},[19,62212],{"className":62213,"style":155},[154],[19,62215,62217],{"className":62216},[159,160,161,162],[19,62218,62220,62223,62226],{"className":62219},[75,162],[19,62221,3297],{"className":62222},[75,76,162],[19,62224,267],{"className":62225},[335,162],[19,62227,110],{"className":62228},[75,162],[19,62230,62231,62234],{"style":31974},[19,62232],{"className":62233,"style":155},[154],[19,62235,62237],{"className":62236},[159,160,161,162],[19,62238,62240],{"className":62239},[75,162],[19,62241,26855],{"className":62242},[75,162],[19,62244,173],{"className":62245},[172],[19,62247,62249],{"className":62248},[142],[19,62250,62252],{"className":62251,"style":31996},[146],[19,62253],{},[19,62255],{"className":62256,"style":412},[411],[19,62258,62260,62303],{"className":62259},[75],[19,62261,62263,62266],{"className":62262},[75],[19,62264,3484],{"className":62265},[75,76],[19,62267,62269],{"className":62268},[133],[19,62270,62272,62295],{"className":62271},[137,138],[19,62273,62275,62292],{"className":62274},[142],[19,62276,62278],{"className":62277,"style":3609},[146],[19,62279,62280,62283],{"style":150},[19,62281],{"className":62282,"style":155},[154],[19,62284,62286],{"className":62285},[159,160,161,162],[19,62287,62289],{"className":62288},[75,162],[19,62290,3297],{"className":62291},[75,76,162],[19,62293,173],{"className":62294},[172],[19,62296,62298],{"className":62297},[142],[19,62299,62301],{"className":62300,"style":180},[146],[19,62302],{},[19,62304,62306,62309],{"className":62305},[75],[19,62307,49],{"className":62308},[75,76],[19,62310,62312],{"className":62311},[133],[19,62313,62315],{"className":62314},[137],[19,62316,62318],{"className":62317},[142],[19,62319,62321],{"className":62320,"style":3429},[146],[19,62322,62323,62326],{"style":1643},[19,62324],{"className":62325,"style":155},[154],[19,62327,62329],{"className":62328},[159,160,161,162],[19,62330,62332],{"className":62331},[75,162],[19,62333,3297],{"className":62334},[75,76,162],[19,62336,62338,62378],{"className":62337,"translate":23},[22],[19,62339,62341],{"className":62340},[27],[29,62342,62343],{"xmlns":31},[33,62344,62345,62375],{},[36,62346,62347,62361],{},[4285,62348,62349,62351,62359],{},[43,62350,26844],{},[36,62352,62353,62355,62357],{},[39,62354,3297],{},[43,62356,267],{},[108,62358,110],{},[39,62360,26855],{"mathvariant":275},[36,62362,62363,62369],{},[103,62364,62365,62367],{},[39,62366,17223],{},[39,62368,3297],{},[269,62370,62371,62373],{},[39,62372,49],{},[39,62374,3297],{},[54,62376,62377],{"encoding":56},"\\sum_{n=0}^{\\infty}{b_{n}x^{n}}",[19,62379,62381],{"className":62380,"ariaHidden":62},[61],[19,62382,62384,62387,62450,62453],{"className":62383},[66],[19,62385],{"className":62386,"style":31929},[70],[19,62388,62390,62393],{"className":62389},[304],[19,62391,26844],{"className":62392,"style":31937},[304,27031,31936],[19,62394,62396],{"className":62395},[133],[19,62397,62399,62442],{"className":62398},[137,138],[19,62400,62402,62439],{"className":62401},[142],[19,62403,62405,62425],{"className":62404,"style":31950},[146],[19,62406,62407,62410],{"style":31953},[19,62408],{"className":62409,"style":155},[154],[19,62411,62413],{"className":62412},[159,160,161,162],[19,62414,62416,62419,62422],{"className":62415},[75,162],[19,62417,3297],{"className":62418},[75,76,162],[19,62420,267],{"className":62421},[335,162],[19,62423,110],{"className":62424},[75,162],[19,62426,62427,62430],{"style":31974},[19,62428],{"className":62429,"style":155},[154],[19,62431,62433],{"className":62432},[159,160,161,162],[19,62434,62436],{"className":62435},[75,162],[19,62437,26855],{"className":62438},[75,162],[19,62440,173],{"className":62441},[172],[19,62443,62445],{"className":62444},[142],[19,62446,62448],{"className":62447,"style":31996},[146],[19,62449],{},[19,62451],{"className":62452,"style":412},[411],[19,62454,62456,62499],{"className":62455},[75],[19,62457,62459,62462],{"className":62458},[75],[19,62460,17223],{"className":62461},[75,76],[19,62463,62465],{"className":62464},[133],[19,62466,62468,62491],{"className":62467},[137,138],[19,62469,62471,62488],{"className":62470},[142],[19,62472,62474],{"className":62473,"style":3609},[146],[19,62475,62476,62479],{"style":150},[19,62477],{"className":62478,"style":155},[154],[19,62480,62482],{"className":62481},[159,160,161,162],[19,62483,62485],{"className":62484},[75,162],[19,62486,3297],{"className":62487},[75,76,162],[19,62489,173],{"className":62490},[172],[19,62492,62494],{"className":62493},[142],[19,62495,62497],{"className":62496,"style":180},[146],[19,62498],{},[19,62500,62502,62505],{"className":62501},[75],[19,62503,49],{"className":62504},[75,76],[19,62506,62508],{"className":62507},[133],[19,62509,62511],{"className":62510},[137],[19,62512,62514],{"className":62513},[142],[19,62515,62517],{"className":62516,"style":3429},[146],[19,62518,62519,62522],{"style":1643},[19,62520],{"className":62521,"style":155},[154],[19,62523,62525],{"className":62524},[159,160,161,162],[19,62526,62528],{"className":62527},[75,162],[19,62529,3297],{"className":62530},[75,76,162]," have the same sum function in a neighborhood of ",[19,62533,62535,62552],{"className":62534,"translate":23},[22],[19,62536,62538],{"className":62537},[27],[29,62539,62540],{"xmlns":31},[33,62541,62542,62550],{},[36,62543,62544,62546,62548],{},[39,62545,49],{},[43,62547,267],{},[108,62549,110],{},[54,62551,51806],{"encoding":56},[19,62553,62555,62573],{"className":62554,"ariaHidden":62},[61],[19,62556,62558,62561,62564,62567,62570],{"className":62557},[66],[19,62559],{"className":62560,"style":3309},[70],[19,62562,49],{"className":62563},[75,76],[19,62565],{"className":62566,"style":613},[411],[19,62568,267],{"className":62569},[335],[19,62571],{"className":62572,"style":613},[411],[19,62574,62576,62579],{"className":62575},[66],[19,62577],{"className":62578,"style":1214},[70],[19,62580,110],{"className":62581},[75],", then these two power series are considered equal in that neighborhood.",[15,62584,62585,62586,62636],{},"Since the coefficients of power series are determined by the sum function and its derivatives at ",[19,62587,62589,62606],{"className":62588,"translate":23},[22],[19,62590,62592],{"className":62591},[27],[29,62593,62594],{"xmlns":31},[33,62595,62596,62604],{},[36,62597,62598,62600,62602],{},[39,62599,49],{},[43,62601,267],{},[108,62603,110],{},[54,62605,51806],{"encoding":56},[19,62607,62609,62627],{"className":62608,"ariaHidden":62},[61],[19,62610,62612,62615,62618,62621,62624],{"className":62611},[66],[19,62613],{"className":62614,"style":3309},[70],[19,62616,49],{"className":62617},[75,76],[19,62619],{"className":62620,"style":613},[411],[19,62622,267],{"className":62623},[335],[19,62625],{"className":62626,"style":613},[411],[19,62628,62630,62633],{"className":62629},[66],[19,62631],{"className":62632,"style":1214},[70],[19,62634,110],{"className":62635},[75],", it follows from the definition of equality of power series that two power series are equal in a neighborhood if and only if their coefficients are equal.",[15,62638,62639,62642,62643,13643,62838,63033,63034,13643,63108,63182],{},[7600,62640,62641],{},"Arithmetic Operations on Power Series:"," Suppose the power series ",[19,62644,62646,62685],{"className":62645,"translate":23},[22],[19,62647,62649],{"className":62648},[27],[29,62650,62651],{"xmlns":31},[33,62652,62653,62683],{},[36,62654,62655,62669],{},[4285,62656,62657,62659,62667],{},[43,62658,26844],{},[36,62660,62661,62663,62665],{},[39,62662,3297],{},[43,62664,267],{},[108,62666,110],{},[39,62668,26855],{"mathvariant":275},[36,62670,62671,62677],{},[103,62672,62673,62675],{},[39,62674,3484],{},[39,62676,3297],{},[269,62678,62679,62681],{},[39,62680,49],{},[39,62682,3297],{},[54,62684,28591],{"encoding":56},[19,62686,62688],{"className":62687,"ariaHidden":62},[61],[19,62689,62691,62694,62757,62760],{"className":62690},[66],[19,62692],{"className":62693,"style":31929},[70],[19,62695,62697,62700],{"className":62696},[304],[19,62698,26844],{"className":62699,"style":31937},[304,27031,31936],[19,62701,62703],{"className":62702},[133],[19,62704,62706,62749],{"className":62705},[137,138],[19,62707,62709,62746],{"className":62708},[142],[19,62710,62712,62732],{"className":62711,"style":31950},[146],[19,62713,62714,62717],{"style":31953},[19,62715],{"className":62716,"style":155},[154],[19,62718,62720],{"className":62719},[159,160,161,162],[19,62721,62723,62726,62729],{"className":62722},[75,162],[19,62724,3297],{"className":62725},[75,76,162],[19,62727,267],{"className":62728},[335,162],[19,62730,110],{"className":62731},[75,162],[19,62733,62734,62737],{"style":31974},[19,62735],{"className":62736,"style":155},[154],[19,62738,62740],{"className":62739},[159,160,161,162],[19,62741,62743],{"className":62742},[75,162],[19,62744,26855],{"className":62745},[75,162],[19,62747,173],{"className":62748},[172],[19,62750,62752],{"className":62751},[142],[19,62753,62755],{"className":62754,"style":31996},[146],[19,62756],{},[19,62758],{"className":62759,"style":412},[411],[19,62761,62763,62806],{"className":62762},[75],[19,62764,62766,62769],{"className":62765},[75],[19,62767,3484],{"className":62768},[75,76],[19,62770,62772],{"className":62771},[133],[19,62773,62775,62798],{"className":62774},[137,138],[19,62776,62778,62795],{"className":62777},[142],[19,62779,62781],{"className":62780,"style":3609},[146],[19,62782,62783,62786],{"style":150},[19,62784],{"className":62785,"style":155},[154],[19,62787,62789],{"className":62788},[159,160,161,162],[19,62790,62792],{"className":62791},[75,162],[19,62793,3297],{"className":62794},[75,76,162],[19,62796,173],{"className":62797},[172],[19,62799,62801],{"className":62800},[142],[19,62802,62804],{"className":62803,"style":180},[146],[19,62805],{},[19,62807,62809,62812],{"className":62808},[75],[19,62810,49],{"className":62811},[75,76],[19,62813,62815],{"className":62814},[133],[19,62816,62818],{"className":62817},[137],[19,62819,62821],{"className":62820},[142],[19,62822,62824],{"className":62823,"style":3429},[146],[19,62825,62826,62829],{"style":1643},[19,62827],{"className":62828,"style":155},[154],[19,62830,62832],{"className":62831},[159,160,161,162],[19,62833,62835],{"className":62834},[75,162],[19,62836,3297],{"className":62837},[75,76,162],[19,62839,62841,62880],{"className":62840,"translate":23},[22],[19,62842,62844],{"className":62843},[27],[29,62845,62846],{"xmlns":31},[33,62847,62848,62878],{},[36,62849,62850,62864],{},[4285,62851,62852,62854,62862],{},[43,62853,26844],{},[36,62855,62856,62858,62860],{},[39,62857,3297],{},[43,62859,267],{},[108,62861,110],{},[39,62863,26855],{"mathvariant":275},[36,62865,62866,62872],{},[103,62867,62868,62870],{},[39,62869,17223],{},[39,62871,3297],{},[269,62873,62874,62876],{},[39,62875,49],{},[39,62877,3297],{},[54,62879,62377],{"encoding":56},[19,62881,62883],{"className":62882,"ariaHidden":62},[61],[19,62884,62886,62889,62952,62955],{"className":62885},[66],[19,62887],{"className":62888,"style":31929},[70],[19,62890,62892,62895],{"className":62891},[304],[19,62893,26844],{"className":62894,"style":31937},[304,27031,31936],[19,62896,62898],{"className":62897},[133],[19,62899,62901,62944],{"className":62900},[137,138],[19,62902,62904,62941],{"className":62903},[142],[19,62905,62907,62927],{"className":62906,"style":31950},[146],[19,62908,62909,62912],{"style":31953},[19,62910],{"className":62911,"style":155},[154],[19,62913,62915],{"className":62914},[159,160,161,162],[19,62916,62918,62921,62924],{"className":62917},[75,162],[19,62919,3297],{"className":62920},[75,76,162],[19,62922,267],{"className":62923},[335,162],[19,62925,110],{"className":62926},[75,162],[19,62928,62929,62932],{"style":31974},[19,62930],{"className":62931,"style":155},[154],[19,62933,62935],{"className":62934},[159,160,161,162],[19,62936,62938],{"className":62937},[75,162],[19,62939,26855],{"className":62940},[75,162],[19,62942,173],{"className":62943},[172],[19,62945,62947],{"className":62946},[142],[19,62948,62950],{"className":62949,"style":31996},[146],[19,62951],{},[19,62953],{"className":62954,"style":412},[411],[19,62956,62958,63001],{"className":62957},[75],[19,62959,62961,62964],{"className":62960},[75],[19,62962,172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have convergence radii ",[19,63035,63037,63055],{"className":63036,"translate":23},[22],[19,63038,63040],{"className":63039},[27],[29,63041,63042],{"xmlns":31},[33,63043,63044,63052],{},[36,63045,63046],{},[103,63047,63048,63050],{},[39,63049,10758],{},[39,63051,3484],{},[54,63053,63054],{"encoding":56},"R_{a}",[19,63056,63058],{"className":63057,"ariaHidden":62},[61],[19,63059,63061,63065],{"className":63060},[66],[19,63062],{"className":63063,"style":63064},[70],"height:0.8333em;vertical-align:-0.15em;",[19,63066,63068,63071],{"className":63067},[75],[19,63069,10758],{"className":63070,"style":10844},[75,76],[19,63072,63074],{"className":63073},[133],[19,63075,63077,63100],{"className":63076},[137,138],[19,63078,63080,63097],{"className":63079},[142],[19,63081,63083],{"className":63082,"style":3609},[146],[19,63084,63085,63088],{"style":10859},[19,63086],{"className":63087,"style":155},[154],[19,63089,63091],{"className":63090},[159,160,161,162],[19,63092,63094],{"className":63093},[75,162],[19,63095,3484],{"className":63096},[75,76,162],[19,63098,173],{"className":63099},[172],[19,63101,63103],{"className":63102},[142],[19,63104,63106],{"className":63105,"style":180},[146],[19,63107],{},[19,63109,63111,63129],{"className":63110,"translate":23},[22],[19,63112,63114],{"className":63113},[27],[29,63115,63116],{"xmlns":31},[33,63117,63118,63126],{},[36,63119,63120],{},[103,63121,63122,63124],{},[39,63123,10758],{},[39,63125,17223],{},[54,63127,63128],{"encoding":56},"R_{b}",[19,63130,63132],{"className":63131,"ariaHidden":62},[61],[19,63133,63135,63138],{"className":63134},[66],[19,63136],{"className":63137,"style":63064},[70],[19,63139,63141,63144],{"className":63140},[75],[19,63142,10758],{"className":63143,"style":10844},[75,76],[19,63145,63147],{"className":63146},[133],[19,63148,63150,63174],{"className":63149},[137,138],[19,63151,63153,63171],{"className":63152},[142],[19,63154,63157],{"className":63155,"style":63156},[146],"height:0.3361em;",[19,63158,63159,63162],{"style":10859},[19,63160],{"className":63161,"style":155},[154],[19,63163,63165],{"className":63164},[159,160,161,162],[19,63166,63168],{"className":63167},[75,162],[19,63169,17223],{"className":63170},[75,76,162],[19,63172,173],{"className":63173},[172],[19,63175,63177],{"className":63176},[142],[19,63178,63180],{"className":63179,"style":180},[146],[19,63181],{},", respectively. Then:",[34217,63184,63185,63675,64411],{},[34220,63186,63187,34262,63225,63674],{},[19,63188,63190,63207],{"className":63189,"translate":23},[22],[19,63191,63193],{"className":63192},[27],[29,63194,63195],{"xmlns":31},[33,63196,63197,63205],{},[36,63198,63199,63201,63203],{},[43,63200,46],{"stretchy":45},[39,63202,34238],{},[43,63204,52],{"stretchy":45},[54,63206,34243],{"encoding":56},[19,63208,63210],{"className":63209,"ariaHidden":62},[61],[19,63211,63213,63216,63219,63222],{"className":63212},[66],[19,63214],{"className":63215,"style":71},[70],[19,63217,46],{"className":63218},[81],[19,63220,34238],{"className":63221},[75,76],[19,63223,52],{"className":63224},[88],[19,63226,63228,63317],{"className":63227,"translate":23},[22],[19,63229,63231],{"className":63230},[27],[29,63232,63233],{"xmlns":31},[33,63234,63235,63314],{},[36,63236,63237,63240,63254,63268,63270,63284,63300,63302,63310,63312],{},[39,63238,63239],{},"λ",[4285,63241,63242,63244,63252],{},[43,63243,26844],{},[36,63245,63246,63248,63250],{},[39,63247,3297],{},[43,63249,267],{},[108,63251,110],{},[39,63253,26855],{"mathvariant":275},[36,63255,63256,63262],{},[103,63257,63258,63260],{},[39,63259,3484],{},[39,63261,3297],{},[269,63263,63264,63266],{},[39,63265,49],{},[39,63267,3297],{},[43,63269,267],{},[4285,63271,63272,63274,63282],{},[43,63273,26844],{},[36,63275,63276,63278,63280],{},[39,63277,3297],{},[43,63279,267],{},[108,63281,110],{},[39,63283,26855],{"mathvariant":275},[36,63285,63286,63288,63294],{},[39,63287,63239],{},[103,63289,63290,63292],{},[39,63291,3484],{},[39,63293,3297],{},[269,63295,63296,63298],{},[39,63297,49],{},[39,63299,3297],{},[43,63301,7830],{"separator":62},[36,63303,63304,63306,63308],{},[43,63305,29161],{"fence":62},[39,63307,49],{},[43,63309,29161],{"fence":62},[43,63311,29168],{},[39,63313,10758],{},[54,63315,63316],{"encoding":56},"\\lambda\\sum_{n=0}^{\\infty}{a_{n}x^{n}}=\\sum_{n=0}^{\\infty}{\\lambda a_{n}x^{n}},\\left| x 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properties can be derived from the corresponding properties of numerical series.",[15,65574,65575],{},"The above discussion regarding power series actually serves as a preliminary study for Taylor series. Although it might seem extensive, it's essential for understanding Taylor series thoroughly.",[15,65577,65578,65579,65622,65623,65674,65675,65747],{},"In the section on Taylor's theorem, we previously mentioned that if a function ",[19,65580,65582,65601],{"className":65581,"translate":23},[22],[19,65583,65585],{"className":65584},[27],[29,65586,65587],{"xmlns":31},[33,65588,65589,65599],{},[36,65590,65591,65593,65595,65597],{},[39,65592,41],{},[43,65594,46],{"stretchy":45},[39,65596,49],{},[43,65598,52],{"stretchy":45},[54,65600,57],{"encoding":56},[19,65602,65604],{"className":65603,"ariaHidden":62},[61],[19,65605,65607,65610,65613,65616,65619],{"className":65606},[66],[19,65608],{"className":65609,"style":71},[70],[19,65611,41],{"className":65612,"style":77},[75,76],[19,65614,46],{"className":65615},[81],[19,65617,49],{"className":65618},[75,76],[19,65620,52],{"className":65621},[88]," has derivatives up to order ",[19,65624,65626,65644],{"className":65625,"translate":23},[22],[19,65627,65629],{"className":65628},[27],[29,65630,65631],{"xmlns":31},[33,65632,65633,65641],{},[36,65634,65635,65637,65639],{},[39,65636,3297],{},[43,65638,2351],{},[108,65640,3495],{},[54,65642,65643],{"encoding":56},"n + 1",[19,65645,65647,65665],{"className":65646,"ariaHidden":62},[61],[19,65648,65650,65653,65656,65659,65662],{"className":65649},[66],[19,65651],{"className":65652,"style":4003},[70],[19,65654,3297],{"className":65655},[75,76],[19,65657],{"className":65658,"style":451},[411],[19,65660,2351],{"className":65661},[455],[19,65663],{"className":65664,"style":451},[411],[19,65666,65668,65671],{"className":65667},[66],[19,65669],{"className":65670,"style":1214},[70],[19,65672,3495],{"className":65673},[75]," in a neighborhood of ",[19,65676,65678,65695],{"className":65677,"translate":23},[22],[19,65679,65681],{"className":65680},[27],[29,65682,65683],{"xmlns":31},[33,65684,65685,65693],{},[36,65686,65687],{},[103,65688,65689,65691],{},[39,65690,49],{},[108,65692,110],{},[54,65694,113],{"encoding":56},[19,65696,65698],{"className":65697,"ariaHidden":62},[61],[19,65699,65701,65704],{"className":65700},[66],[19,65702],{"className":65703,"style":123},[70],[19,65705,65707,65710],{"className":65706},[75],[19,65708,49],{"className":65709},[75,76],[19,65711,65713],{"className":65712},[133],[19,65714,65716,65739],{"className":65715},[137,138],[19,65717,65719,65736],{"className":65718},[142],[19,65720,65722],{"className":65721,"style":147},[146],[19,65723,65724,65727],{"style":150},[19,65725],{"className":65726,"style":155},[154],[19,65728,65730],{"className":65729},[159,160,161,162],[19,65731,65733],{"className":65732},[75,162],[19,65734,110],{"className":65735},[75,162],[19,65737,173],{"className":65738},[172],[19,65740,65742],{"className":65741},[142],[19,65743,65745],{"className":65744,"style":180},[146],[19,65746],{},", then the Taylor series expansion is given 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= f(x_{0}) + f'(x_{0})(x-x_{0}) + \\frac{f''(x_{0})}{2!}(x-x_{0})^{2} + \\ldots + \\frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n} + R_{n}(x)",[19,65941,65943,65970,66037,66139,66200,66364,66454,66472,66642,66732],{"className":65942,"ariaHidden":62},[61],[19,65944,65946,65949,65952,65955,65958,65961,65964,65967],{"className":65945},[66],[19,65947],{"className":65948,"style":71},[70],[19,65950,41],{"className":65951,"style":77},[75,76],[19,65953,46],{"className":65954},[81],[19,65956,49],{"className":65957},[75,76],[19,65959,52],{"className":65960},[88],[19,65962],{"className":65963,"style":613},[411],[19,65965,267],{"className":65966},[335],[19,65968],{"className":65969,"style":613},[411],[19,65971,65973,65976,65979,65982,66025,66028,66031,66034],{"className":65972},[66],[19,65974],{"className":65975,"style":71},[70],[19,65977,41],{"className":65978,"style":77},[75,76],[19,65980,46],{"className":65981},[81],[19,65983,65985,65988],{"className":65984},[75],[19,65986,49],{"className":65987},[75,76],[19,65989,65991],{"className":65990},[133],[19,65992,65994,66017],{"className":65993},[137,138],[19,65995,65997,66014],{"className":65996},[142],[19,65998,66000],{"className":65999,"style":147},[146],[19,66001,66002,66005],{"style":150},[19,66003],{"className":66004,"style":155},[154],[19,66006,66008],{"className":66007},[159,160,161,162],[19,66009,66011],{"className":66010},[75,162],[19,66012,110],{"className":66013},[75,162],[19,66015,173],{"className":66016},[172],[19,66018,66020],{"className":66019},[142],[19,66021,66023],{"className":66022,"style":180},[146],[19,66024],{},[19,66026,52],{"className":66027},[88],[19,66029],{"className":66030,"style":451},[411],[19,66032,2351],{"className":66033},[455],[19,66035],{"className":66036,"style":451},[411],[19,66038,66040,66043,66075,66078,66121,66124,66127,66130,66133,66136],{"className":66039},[66],[19,66041],{"className":66042,"style":626},[70],[19,66044,66046,66049],{"className":66045},[75],[19,66047,41],{"className":66048,"style":77},[75,76],[19,66050,66052],{"className":66051},[133],[19,66053,66055],{"className":66054},[137],[19,66056,66058],{"className":66057},[142],[19,66059,66061],{"className":66060,"style":645},[146],[19,66062,66063,66066],{"style":648},[19,66064],{"className":66065,"style":155},[154],[19,66067,66069],{"className":66068},[159,160,161,162],[19,66070,66072],{"className":66071},[75,162],[19,66073,277],{"className":66074},[75,162],[19,66076,46],{"className":66077},[81],[19,66079,66081,66084],{"className":66080},[75],[19,66082,49],{"className":66083},[75,76],[19,66085,66087],{"className":66086},[133],[19,66088,66090,66113],{"className":66089},[137,138],[19,66091,66093,66110],{"className":66092},[142],[19,66094,66096],{"className":66095,"style":147},[146],[19,66097,66098,66101],{"style":150},[19,66099],{"className":66100,"style":155},[154],[19,66102,66104],{"className":66103},[159,160,161,162],[19,66105,66107],{"className":66106},[75,162],[19,66108,110],{"className":66109},[75,162],[19,66111,173],{"className":66112},[172],[19,66114,66116],{"className":66115},[142],[19,66117,66119],{"className":66118,"style":180},[146],[19,66120],{},[19,66122,52],{"className":66123},[88],[19,66125,46],{"className":66126},[81],[19,66128,49],{"className":66129},[75,76],[19,66131],{"className":66132,"style":451},[411],[19,66134,240],{"className":66135},[455],[19,66137],{"className":66138,"style":451},[411],[19,66140,66142,66145,66188,66191,66194,66197],{"className":66141},[66],[19,66143],{"className":66144,"style":71},[70],[19,66146,66148,66151],{"className":66147},[75],[19,66149,49],{"className":66150},[75,76],[19,66152,66154],{"className":66153},[133],[19,66155,66157,66180],{"className":66156},[137,138],[19,66158,66160,66177],{"className":66159},[142],[19,66161,66163],{"className":66162,"style":147},[146],[19,66164,66165,66168],{"style":150},[19,66166],{"className":66167,"style":155},[154],[19,66169,66171],{"className":66170},[159,160,161,162],[19,66172,66174],{"className":66173},[75,162],[19,66175,110],{"className":66176},[75,162],[19,66178,173],{"className":66179},[172],[19,66181,66183],{"className":66182},[142],[19,66184,66186],{"className":66185,"style":180},[146],[19,66187],{},[19,66189,52],{"className":66190},[88],[19,66192],{"className":66193,"style":451},[411],[19,66195,2351],{"className":66196},[455],[19,66198],{"className":66199,"style":451},[411],[19,66201,66203,66206,66349,66352,66355,66358,66361],{"className":66202},[66],[19,66204],{"className":66205,"style":5507},[70],[19,66207,66209,66212,66346],{"className":66208},[75],[19,66210],{"className":66211},[81,422],[19,66213,66215],{"className":66214},[226],[19,66216,66218,66338],{"className":66217},[137,138],[19,66219,66221,66335],{"className":66220},[142],[19,66222,66224,66238,66246],{"className":66223,"style":1460},[146],[19,66225,66226,66229],{"style":438},[19,66227],{"className":66228,"style":322},[154],[19,66230,66232,66235],{"className":66231},[75],[19,66233,3518],{"className":66234},[75],[19,66236,4341],{"className":66237},[88],[19,66239,66240,66243],{"style":504},[19,66241],{"className":66242,"style":322},[154],[19,66244],{"classN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= 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ssName":67185,"style":155},[154],[19,67187,67189],{"className":67188},[159,160,161,162],[19,67190,67192,67195,67198],{"className":67191},[75,162],[19,67193,3297],{"className":67194},[75,76,162],[19,67196,2351],{"className":67197},[455,162],[19,67199,3495],{"className":67200},[75,162],[19,67202,67204,67218],{"className":67203,"translate":23},[22],[19,67205,67207],{"className":67206},[27],[29,67208,67209],{"xmlns":31},[33,67210,67211,67215],{},[36,67212,67213],{},[39,67214,17519],{},[54,67216,67217],{"encoding":56},"\\xi",[19,67219,67221],{"className":67220,"ariaHidden":62},[61],[19,67222,67224,67227],{"className":67223},[66],[19,67225],{"className":67226,"style":6202},[70],[19,67228,17519],{"className":67229,"style":17547},[75,76]," lies between ",[19,67232,67234,67247],{"className":67233,"translate":23},[22],[19,67235,67237],{"className":67236},[27],[29,67238,67239],{"xmlns":31},[33,67240,67241,67245],{},[36,67242,67243],{},[39,67244,49],{},[54,67246,49],{"encoding":56},[19,67248,67250],{"className":67249,"ariaHidden":62},[61],[19,67251,67253,67256],{"className":67252},[66],[19,67254],{"className":67255,"style":3309},[70],[19,67257,49],{"className":67258},[75,76],[19,67260,67262,67279],{"className":67261,"translate":23},[22],[19,67263,67265],{"className":67264},[27],[29,67266,67267],{"xmlns":31},[33,67268,67269,67277],{},[36,67270,67271],{},[103,67272,67273,67275],{},[39,67274,49],{},[108,67276,110],{},[54,67278,113],{"encoding":56},[19,67280,67282],{"className":67281,"ariaHidden":62},[61],[19,67283,67285,67288],{"className":67284},[66],[19,67286],{"className":67287,"style":123},[70],[19,67289,67291,67294],{"className":67290},[75],[19,67292,49],{"className":67293},[75,76],[19,67295,67297],{"className":67296},[133],[19,67298,67300,67323],{"className":67299},[137,138],[19,67301,67303,67320],{"className":67302},[142],[19,67304,67306],{"className":67305,"style":147},[146],[19,67307,67308,67311],{"style":150},[19,67309],{"className":67310,"style":155},[154],[19,67312,67314],{"className":67313},[159,160,161,162],[19,67315,67317],{"className":67316},[75,162],[19,67318,110],{"className":67319},[75,162],[19,67321,173],{"className":67322},[172],[19,67324,67326],{"className":67325},[142],[19,67327,67329],{"className":67328,"style":180},[146],[19,67330],{},[15,67332,67333,67334,67377,67378,67473],{},"Inspired by the form of Taylor's theorem, if a function ",[19,67335,67337,67356],{"className":67336,"translate":23},[22],[19,67338,67340],{"className":67339},[27],[29,67341,67342],{"xmlns":31},[33,67343,67344,67354],{},[36,67345,67346,67348,67350,67352],{},[39,67347,41],{},[43,67349,46],{"stretchy":45},[39,67351,49],{},[43,67353,52],{"stretchy":45},[54,67355,57],{"encoding":56},[19,67357,67359],{"className":67358,"ariaHidden":62},[61],[19,67360,67362,67365,67368,67371,67374],{"className":67361},[66],[19,67363],{"className":67364,"style":71},[70],[19,67366,41],{"className":67367,"style":77},[75,76],[19,67369,46],{"className":67370},[81],[19,67372,49],{"className":67373},[75,76],[19,67375,52],{"className":67376},[88]," has derivatives of all orders at ",[19,67379,67381,67403],{"className":67380,"translate":23},[22],[19,67382,67384],{"className":67383},[27],[29,67385,67386],{"xmlns":31},[33,67387,67388,67400],{},[36,67389,67390,67392,67394],{},[39,67391,49],{},[43,67393,267],{},[103,67395,67396,67398],{},[39,67397,49],{},[108,67399,110],{},[54,67401,67402],{"encoding":56},"x = x_{0}",[19,67404,67406,67424],{"className":67405,"ariaHidden":62},[61],[19,67407,67409,67412,67415,67418,67421],{"className":67408},[66],[19,67410],{"className":67411,"style":3309},[70],[19,67413,49],{"className":67414},[75,76],[19,67416],{"className":67417,"style":613},[411],[19,67419,267],{"className":67420},[335],[19,67422],{"className":67423,"style":613},[411],[19,67425,67427,67430],{"className":67426},[66],[19,67428],{"className":67429,"style":123},[70],[19,67431,67433,67436],{"className":67432},[75],[19,67434,49],{"className":67435},[75,76],[19,67437,67439],{"className":67438},[133],[19,67440,67442,67465],{"className":67441},[137,138],[19,67443,67445,67462],{"className":67444},[142],[19,67446,67448],{"className":67447,"style":147},[146],[19,67449,67450,67453],{"style":150},[19,67451],{"className":67452,"style":155},[154],[19,67454,67456],{"className":67455},[159,160,161,162],[19,67457,67459],{"className":67458},[75,162],[19,67460,110],{"className":67461},[75,162],[19,67463,173],{"className":67464},[172],[19,67466,67468],{"className":67467},[142],[19,67469,67471],{"className":67470,"style":180},[146],[19,67472],{},", then we can construct a power series:",[19,67475,67477],{"className":67476,"translate":23},[187],[19,67478,67480,67636],{"className":67479,"translate":23},[22],[19,67481,67483],{"className":67482},[27],[29,67484,67485],{"xmlns":31,"display":196},[33,67486,67487,67633],{},[36,67488,67489,67491,67493,67499,67501,67503,67509,67511,67517,67519,67521,67523,67525,67531,67533,67535,67565,67567,67569,67571,67577,67583,67585,67587,67589,67629,67631],{},[39,67490,41],{},[43,67492,46],{"stretchy":45},[103,67494,67495,67497],{},[39,67496,49],{},[108,67498,110],{},[43,67500,52],{"stretchy":45},[43,67502,2351],{},[269,67504,67505,67507],{},[39,67506,41],{},[43,67508,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,67510,46],{"stretchy":45},[103,67512,67513,67515],{},[39,67514,49],{},[108,67516,110],{},[43,67518,52],{"stretchy":45},[43,67520,46],{"stretchy":45},[39,67522,49],{},[43,67524,240],{},[103,67526,67527,67529],{},[39,67528,49],{},[108,67530,110],{},[43,67532,52],{"stretchy":45},[43,67534,2351],{},[226,67536,67537,67559],{},[36,67538,67539,67549,67551,67557],{},[269,67540,67541,67543],{},[39,67542,41],{},[36,67544,67545,67547],{},[43,67546,277],{"mathvariant":275},[43,67548,277],{"mathvariant":275},[43,67550,46],{"stretchy":45},[103,67552,67553,67555],{},[39,67554,49],{},[108,67556,110],{},[43,67558,52],{"stretchy":45},[36,67560,67561,67563],{},[108,67562,3518],{},[43,67564,4341],{"stretchy":45},[43,67566,46],{"stretchy":45},[39,67568,49],{},[43,67570,240],{},[103,67572,67573,67575],{},[39,67574,49],{},[108,67576,110],{},[269,67578,67579,67581],{},[43,67580,52],{"stretchy":45},[108,67582,3518],{},[43,67584,2351],{},[43,67586,21943],{},[43,67588,2351],{},[226,67590,67591,67623],{},[36,67592,67593,67605,67607,67609,67611,67617],{},[269,67594,67595,67597],{},[39,67596,41],{},[36,67598,67599,67601,67603],{},[43,67600,46],{"stretchy":45},[39,67602,3297],{},[43,67604,52],{"stretchy":45},[43,67606,46],{"stretchy":45},[39,67608,49],{},[43,67610,240],{},[103,67612,67613,67615],{},[39,67614,49],{},[108,67616,110],{},[269,67618,67619,67621],{},[43,67620,52],{"stretchy":45},[39,67622,3297],{},[36,67624,67625,67627],{},[39,67626,3297],{},[43,67628,4341],{"stretchy":45},[43,67630,2351],{},[43,67632,21943],{},[54,67634,67635],{"encoding":56},"f(x_{0}) + f'(x_{0})(x-x_{0}) + \\frac{f''(x_{0})}{2!}(x-x_{0})^{2} + \\ldots + \\frac{f^{(n)}(x-x_{0})^{n}}{n!} + \\ldots",[19,67637,67639,67706,67808,67869,68033,68123,68141,68346],{"className":67638,"ariaHidden":62},[61],[19,67640,67642,67645,67648,67651,67694,67697,67700,67703],{"className":67641},[66],[19,67643],{"className":67644,"style":71},[70],[19,67646,41],{"className":67647,"style":77},[75,76],[19,67649,46],{"className":67650},[81],[19,67652,67654,67657],{"className":67653},[75],[19,67655,49],{"className":67656},[75,76],[19,67658,67660],{"className":67659},[133],[19,67661,67663,67686],{"className":67662},[137,138],[19,67664,67666,67683],{"className":67665},[142],[19,67667,67669],{"className":67668,"style":147},[146],[19,67670,67671,67674],{"style":150},[19,67672],{"className":67673,"style":155},[154],[19,67675,67677],{"className":67676},[159,160,161,162],[19,67678,67680],{"className":67679},[75,162],[19,67681,110],{"className":67682},[75,162],[19,67684,173],{"className":67685},[172],[19,67687,67689],{"className":67688},[142],[19,67690,67692],{"className":67691,"style":180},[146],[19,67693],{},[19,67695,52],{"className":67696},[88],[19,67698],{"className":67699,"style":451},[411],[19,67701,2351],{"className":67702},[455],[19,67704],{"className":67705,"style":451},[411],[19,67707,67709,67712,67744,67747,67790,67793,67796,67799,67802,67805],{"className":67708},[66],[19,67710],{"className":67711,"style":626},[70],[19,67713,67715,67718],{"className":67714},[75],[19,67716,41],{"className":67717,"style":77},[75,76],[19,67719,67721],{"className":67720},[133],[19,67722,67724],{"className":67723},[137],[19,67725,67727],{"className":67726},[142],[19,67728,67730],{"className":67729,"style":645},[146],[19,67731,67732,67735],{"style":648},[19,67733],{"className":67734,"style":155},[154],[19,67736,67738],{"className":67737},[159,160,161,162],[19,67739,67741],{"className":67740},[75,162],[19,67742,277],{"className":67743},[75,162],[19,67745,46],{"className":67746},[81],[19,67748,67750,67753],{"className":67749},[75],[19,67751,49],{"className":67752},[75,76],[19,67754,67756],{"className":67755},[133],[19,67757,67759,67782],{"className":67758},[137,138],[19,67760,67762,67779],{"className":67761},[142],[19,67763,67765],{"className":67764,"style":147},[146],[19,67766,67767,67770],{"style":150},[19,67768],{"className":67769,"style":155},[154],[19,67771,67773],{"className":67772},[159,160,161,162],[19,67774,67776],{"className":67775},[75,162],[19,67777,110],{"className":67778},[75,162],[19,67780,173],{"className":67781},[172],[19,67783,67785],{"className":67784},[142],[19,67786,67788],{"className":67787,"style":180},[146],[19,67789],{},[19,67791,52],{"className":67792},[88],[19,67794,46],{"className":67795},[81],[19,67797,49],{"className":67798},[75,76],[19,67800],{"className":67801,"style":451},[411],[19,67803,240],{"className":67804},[455],[19,67806],{"className":67807,"style":451},[411],[19,67809,67811,67814,67857,67860,67863,67866],{"className":67810},[66],[19,67812],{"className":67813,"style":71},[70],[19,67815,67817,67820],{"className":67816},[75],[19,67818,49],{"className":67819},[75,76],[19,67821,67823],{"className":67822},[133],[19,67824,67826,67849],{"className":67825},[137,138],[19,67827,67829,67846],{"className":67828},[142],[19,67830,67832],{"className":67831,"style":147},[146],[19,67833,67834,67837],{"style":150},[19,67835],{"className":67836,"style":155},[154],[19,67838,67840],{"className":67839},[159,160,161,162],[19,67841,67843],{"className":67842},[75,162],[19,67844,110],{"className":67845},[75,162],[19,67847,173],{"className":67848},[172],[19,67850,67852],{"className":67851},[142],[19,67853,67855],{"className":67854,"style":180},[146],[19,67856],{},[19,67858,52],{"className":67859},[88],[19,67861],{"className":67862,"style":451},[411],[19,67864,2351],{"className":67865},[455],[19,67867],{"className":67868,"style":451},[411],[19,67870,67872,67875,68018,68021,68024,68027,68030],{"className":67871},[66],[19,67873],{"className":67874,"style":5507},[70],[19,67876,67878,67881,68015],{"className":67877},[75],[19,67879],{"className":67880},[81,422],[19,67882,67884],{"className":67883},[226],[19,67885,67887,68007],{"className":67886},[137,138],[19,67888,67890,68004],{"className":67889},[142],[19,67891,67893,67907,67915],{"className":67892,"style":1460},[146],[19,67894,67895,67898],{"style":438},[19,67896],{"className":67897,"style":322},[154],[19,67899,67901,67904],{"className":67900},[75],[19,67902,3518],{"className":67903},[75],[19,67905,4341],{"className":67906},[88],[19,67908,67909,67912],{"style":504},[19,67910],{"className":67911,"style":322},[154],[19,67913],{"className":67914,"style":512},[511],[19,67916,67917,67920],{"style":515},[19,67918],{"className":67919,"style":322},[154],[19,67921,67923,67955,67958,68001],{"className":67922},[75],[19,67924,67926,67929],{"className":67925},[75],[19,67927,41],{"className":67928,"style":77},[75,76],[19,67930,67932],{"className":67931},[133],[19,67933,67935],{"className":67934},[137],[19,67936,67938],{"className":67937},[142],[19,67939,67941],{"className":67940,"style":1640},[146],[19,67942,67943,67946],{"style":1643},[19,67944],{"className":67945,"style":155},[154],[19,67947,67949],{"className":67948},[159,160,161,162],[19,67950,67952],{"className":67951},[75,162],[19,67953,4786],{"className":67954},[75,162],[19,67956,46],{"className":67957},[81],[19,67959,67961,67964],{"className":67960},[75],[19,67962,49],{"className":67963},[75,76],[19,67965,67967],{"className":67966},[133],[19,67968,67970,67993],{"className":67969},[137,138],[19,67971,67973,67990],{"className":67972},[142],[19,67974,67976],{"className":67975,"style":147},[146],[19,67977,67978,67981],{"style":150},[19,67979],{"className":67980,"style":155},[154],[19,67982,67984],{"className":67983},[159,160,161,162],[19,67985,67987],{"className":67986},[75,162],[19,67988,110],{"className":67989},[75,162],[19,67991,173],{"className":67992},[172],[19,67994,67996],{"className":67995},[142],[19,67997,67999],{"className":67998,"style":180},[146],[19,68000],{},[19,68002,52],{"className":68003},[88],[19,68005,173],{"className":68006},[172],[19,68008,68010],{"className":68009},[142],[19,68011,68013],{"className":68012,"style":5671},[146],[19,68014],{},[19,68016],{"className":68017},[88,422],[19,68019,46],{"className":68020},[81],[19,68022,49],{"className":68023},[75,76],[19,68025],{"className":68026,"style":451},[411],[19,68028,240],{"className":68029},[455],[19,68031],{"className":68032,"style":451},[411],[19,68034,68036,68039,68082,68114,68117,68120],{"className":68035},[66],[19,68037],{"className":68038,"style":3908},[70],[19,68040,68042,68045],{"className":68041},[75],[19,68043,49],{"className":68044},[75,76],[19,68046,68048],{"className":68047},[133],[19,68049,68051,68074],{"className":68050},[137,138],[19,68052,68054,68071],{"className":68053},[142],[19,68055,68057],{"className":68056,"style":147},[146],[19,68058,68059,68062],{"style":150},[19,68060],{"className":68061,"style":155},[154],[19,68063,68065],{"className":68064},[159,160,161,162],[19,68066,68068],{"className":68067},[75,162],[19,68069,110],{"className":68070},[75,162],[19,68072,173],{"className":68073},[172],[19,68075,68077],{"className":68076},[142],[19,68078,68080],{"className":68079,"style":180},[146],[19,68081],{},[19,68083,68085,68088],{"className":68084},[88],[19,68086,52],{"className":68087},[88],[19,68089,68091],{"className":68090},[133],[19,68092,68094],{"className":68093},[137],[19,68095,68097],{"className":68096},[142],[19,68098,68100],{"className":68099,"style":3970},[146],[19,68101,68102,68105],{"style":648},[19,68103],{"className":68104,"style":155},[154],[19,68106,68108],{"className":68107},[159,160,161,162],[19,68109,68111],{"className":68110},[75,162],[19,68112,3518],{"className":68113},[75,162],[19,68115],{"className":68116,"style":451},[411],[19,68118,2351],{"className":68119},[455],[19,68121],{"className":68122,"style":451},[411],[19,68124,68126,68129,68132,68135,68138],{"className":68125},[66],[19,68127],{"className":68128,"style":4003},[70],[19,68130,21943],{"className":68131},[16026],[19,68133],{"className":68134,"style":451},[411],[19,68136,2351],{"className":68137},[455],[19,68139],{"className":68140,"style":451},[411],[19,68142,68144,68147,68337,68340,68343],{"className":68143},[66],[19,68145],{"className":68146,"style":7348},[70],[19,68148,68150,68153,68334],{"className":68149},[75],[19,68151],{"className":68152},[81,422],[19,68154,68156],{"className":68155},[226],[19,68157,68159,68326],{"className":68158},[137,138],[19,68160,68162,68323],{"className":68161},[142],[19,68163,68165,68179,68187],{"className":68164,"style":7367},[146],[19,68166,68167,68170],{"style":438},[19,68168],{"className":68169,"style":322},[154],[19,68171,68173,68176],{"className":68172},[75],[19,68174,3297],{"className":68175},[75,76],[19,68177,4341],{"className":68178},[88],[19,68180,68181,68184],{"style":504},[19,68182],{"className":68183,"style":322},[154],[19,68185],{"className":68186,"style":512},[511],[19,68188,68189,68192],{"style":515},[19,68190],{"className":68191,"style":322},[154],[19,68193,68195,68233,68236,68239,68242,68245,68248,68291],{"className":68194},[75],[19,68196,68198,68201],{"className":68197},[75],[19,68199,41],{"className":68200,"style":77},[75,76],[19,68202,68204],{"className":68203},[133],[19,68205,68207],{"className":68206},[137],[19,68208,68210],{"className":68209},[142],[19,68211,68213],{"className":68212,"style":7416},[146],[19,68214,68215,68218],{"style":1643},[19,68216],{"className":68217,"style":155},[154],[19,68219,68221],{"className":68220},[159,160,161,162],[19,68222,68224,68227,68230],{"className":68223},[75,162],[19,68225,46],{"className":68226},[81,162],[19,68228,3297],{"className":68229},[75,76,162],[19,68231,52],{"className":68232},[88,162],[19,68234,46],{"className":68235},[81],[19,68237,49],{"className":68238},[75,76],[19,68240],{"className":68241,"style":451},[411],[19,68243,240],{"className":68244},[455],[19,68246],{"className":68247,"style":451},[411],[19,68249,68251,68254],{"className":68250},[75],[19,68252,49],{"className":68253},[75,76],[19,68255,68257],{"className":68256},[133],[19,68258,68260,68283],{"className":68259},[137,138],[19,68261,68263,68280],{"className":68262},[142],[19,68264,68266],{"className":68265,"style":147},[146],[19,68267,68268,68271],{"style":150},[19,68269],{"className":68270,"style":155},[154],[19,68272,68274],{"className":68273},[159,160,161,162],[19,68275,68277],{"className":68276},[75,162],[19,68278,110],{"className":68279},[75,162],[19,68281,173],{"className":68282},[172],[19,68284,68286],{"className":68285},[142],[19,68287,68289],{"className":68288,"style":180},[146],[19,68290],{},[19,68292,68294,68297],{"className":68293},[88],[19,68295,52],{"className":68296},[88],[19,68298,68300],{"className":68299},[133],[19,68301,68303],{"className":68302},[137],[19,68304,68306],{"className":68305},[142],[19,68307,68309],{"className":68308,"style":3429},[146],[19,68310,68311,68314],{"style":1643},[19,68312],{"className":68313,"style":155},[154],[19,68315,68317],{"className":68316},[159,160,161,162],[19,68318,68320],{"className":68319},[75,162],[19,68321,3297],{"className":68322},[75,76,162],[19,68324,173],{"className":68325},[172],[19,68327,68329],{"className":68328},[142],[19,68330,68332],{"className":68331,"style":5671},[146],[19,68333],{},[19,68335],{"className":68336},[88,422],[19,68338],{"className":68339,"style":451},[411],[19,68341,2351],{"className":68342},[455],[19,68344],{"className":68345,"style":451},[411],[19,68347,68349,68352],{"className":68348},[66],[19,68350],{"className":68351,"style":23412},[70],[19,68353,21943],{"className":68354},[16026],[15,68356,68357,68358,68361,68362,68405,68406,68478,68479,68522,68523,68595,68596,68639,68640,68743,68744,68845,68846,68889,68890,69086],{},"This series is called the ",[7600,68359,68360],{},"Taylor series"," of the function ",[19,68363,68365,68384],{"className":68364,"translate":23},[22],[19,68366,68368],{"className":68367},[27],[29,68369,68370],{"xmlns":31},[33,68371,68372,68382],{},[36,68373,68374,68376,68378,68380],{},[39,68375,41],{},[43,68377,46],{"stretchy":45},[39,68379,49],{},[43,68381,52],{"stretchy":45},[54,68383,57],{"encoding":56},[19,68385,68387],{"className":68386,"ariaHidden":62},[61],[19,68388,68390,68393,68396,68399,68402],{"className":68389},[66],[19,68391],{"className":68392,"style":71},[70],[19,68394,41],{"className":68395,"style":77},[75,76],[19,68397,46],{"className":68398},[81],[19,68400,49],{"className":68401},[75,76],[19,68403,52],{"className":68404},[88]," at the point ",[19,68407,68409,68426],{"className":68408,"translate":23},[22],[19,68410,68412],{"className":68411},[27],[29,68413,68414],{"xmlns":31},[33,68415,68416,68424],{},[36,68417,68418],{},[103,68419,68420,68422],{},[39,68421,49],{},[108,68423,110],{},[54,68425,113],{"encoding":56},[19,68427,68429],{"className":68428,"ariaHidden":62},[61],[19,68430,68432,68435],{"className":68431},[66],[19,68433],{"className":68434,"style":123},[70],[19,68436,68438,68441],{"className":68437},[75],[19,68439,49],{"className":68440},[75,76],[19,68442,68444],{"className":68443},[133],[19,68445,68447,68470],{"className":68446},[137,138],[19,68448,68450,68467],{"className":68449},[142],[19,68451,68453],{"className":68452,"style":147},[146],[19,68454,68455,68458],{"style":150},[19,68456],{"className":68457,"style":155},[154],[19,68459,68461],{"className":68460},[159,160,161,162],[19,68462,68464],{"className":68463},[75,162],[19,68465,110],{"className":68466},[75,162],[19,68468,173],{"className":68469},[172],[19,68471,68473],{"className":68472},[142],[19,68474,68476],{"className":68475,"style":180},[146],[19,68477],{},". Now, we face the question: What is the relationship between the function ",[19,68480,68482,68501],{"className":68481,"translate":23},[22],[19,68483,68485],{"className":68484},[27],[29,68486,68487],{"xmlns":31},[33,68488,68489,68499],{},[36,68490,68491,68493,68495,68497],{},[39,68492,41],{},[43,68494,46],{"stretchy":45},[39,68496,49],{},[43,68498,52],{"stretchy":45},[54,68500,57],{"encoding":56},[19,68502,68504],{"className":68503,"ariaHidden":62},[61],[19,68505,68507,68510,68513,68516,68519],{"className":68506},[66],[19,68508],{"className":68509,"style":71},[70],[19,68511,41],{"className":68512,"style":77},[75,76],[19,68514,46],{"className":68515},[81],[19,68517,49],{"className":68518},[75,76],[19,68520,52],{"className":68521},[88]," and its Taylor series expansion at ",[19,68524,68526,68543],{"className":68525,"translate":23},[22],[19,68527,68529],{"className":68528},[27],[29,68530,68531],{"xmlns":31},[33,68532,68533,68541],{},[36,68534,68535],{},[103,68536,68537,68539],{},[39,68538,49],{},[108,68540,110],{},[54,68542,113],{"encoding":56},[19,68544,68546],{"className":68545,"ariaHidden":62},[61],[19,68547,68549,68552],{"className":68548},[66],[19,68550],{"className":68551,"style":123},[70],[19,68553,68555,68558],{"className":68554},[75],[19,68556,49],{"className":68557},[75,76],[19,68559,68561],{"className":68560},[133],[19,68562,68564,68587],{"className":68563},[137,138],[19,68565,68567,68584],{"className":68566},[142],[19,68568,68570],{"className":68569,"style":147},[146],[19,68571,68572,68575],{"style":150},[19,68573],{"className":68574,"style":155},[154],[19,68576,68578],{"className":68577},[159,160,161,162],[19,68579,68581],{"className":68580},[75,162],[19,68582,110],{"className":68583},[75,162],[19,68585,173],{"className":68586},[172],[19,68588,68590],{"className":68589},[142],[19,68591,68593],{"className":68592,"style":180},[146],[19,68594],{},"? If the function ",[19,68597,68599,68618],{"className":68598,"translate":23},[22],[19,68600,68602],{"className":68601},[27],[29,68603,68604],{"xmlns":31},[33,68605,68606,68616],{},[36,68607,68608,68610,68612,68614],{},[39,68609,41],{},[43,68611,46],{"stretchy":45},[39,68613,49],{},[43,68615,52],{"stretchy":45},[54,68617,57],{"encoding":56},[19,68619,68621],{"className":68620,"ariaHidden":62},[61],[19,68622,68624,68627,68630,68633,68636],{"className":68623},[66],[19,68625],{"className":68626,"style":71},[70],[19,68628,41],{"className":68629,"style":77},[75,76],[19,68631,46],{"className":68632},[81],[19,68634,49],{"className":68635},[75,76],[19,68637,52],{"className":68638},[88]," is defined in a neighborhood ",[19,68641,68643,68672],{"className":68642,"translate":23},[22],[19,68644,68646],{"className":68645},[27],[29,68647,68648],{"xmlns":31},[33,68649,68650,68669],{},[36,68651,68652,68654,68656,68662,68664,68667],{},[39,68653,13196],{},[43,68655,46],{"stretchy":45},[103,68657,68658,68660],{},[39,68659,49],{},[108,68661,110],{},[43,68663,7830],{"separator":62},[39,68665,68666],{},"δ",[43,68668,52],{"stretchy":45},[54,68670,68671],{"encoding":56},"U(x_{0},\\delta)",[19,68673,68675],{"className":68674,"ariaHidden":62},[61],[19,68676,68678,68681,68684,68687,68730,68733,68736,68740],{"className":68677},[66],[19,68679],{"className":68680,"style":71},[70],[19,68682,13196],{"className":68683,"style":13222},[75,76],[19,68685,46],{"className":68686},[81],[19,68688,68690,68693],{"className":68689},[75],[19,68691,49],{"className":68692},[75,76],[19,68694,68696],{"className":68695},[133],[19,68697,68699,68722],{"className":68698},[137,138],[19,68700,68702,68719],{"className":68701},[142],[19,68703,68705],{"className":68704,"style":147},[146],[19,68706,68707,68710],{"style":150},[19,68708],{"className":68709,"style":155},[154],[19,68711,68713],{"className":68712},[159,160,161,162],[19,68714,68716],{"className":68715},[75,162],[19,68717,110],{"className":68718},[75,162],[19,68720,173],{"className":68721},[172],[19,68723,68725],{"className":68724},[142],[19,68726,68728],{"className":68727,"style":180},[146],[19,68729],{},[19,68731,7830],{"className":68732},[8036],[19,68734],{"className":68735,"style":412},[411],[19,68737,68666],{"className":68738,"style":68739},[75,76],"margin-right:0.03785em;",[19,68741,52],{"className":68742},[88]," and its Taylor series converges in the interval 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equals the sum function of its Taylor series on the interval 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if and only if, for all 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represents the Lagrange remainder term of ",[19,69876,69878,69897],{"className":69877,"translate":23},[22],[19,69879,69881],{"className":69880},[27],[29,69882,69883],{"xmlns":31},[33,69884,69885,69895],{},[36,69886,69887,69889,69891,69893],{},[39,69888,41],{},[43,69890,46],{"stretchy":45},[39,69892,49],{},[43,69894,52],{"stretchy":45},[54,69896,57],{"encoding":56},[19,69898,69900],{"className":69899,"ariaHidden":62},[61],[19,69901,69903,69906,69909,69912,69915],{"className":69902},[66],[19,69904],{"className":69905,"style":71},[70],[19,69907,41],{"className":69908,"style":77},[75,76],[19,69910,46],{"className":69911},[81],[19,69913,49],{"className":69914},[75,76],[19,69916,52],{"className":69917},[88],[19,69919,69921,69938],{"className":69920,"translate":23},[22],[19,69922,69924],{"className":69923},[27],[29,69925,69926],{"xmlns":31},[33,69927,69928,69936],{},[36,69929,69930],{},[103,69931,69932,69934],{},[39,69933,49],{},[108,69935,110],{},[54,69937,113],{"encoding":56},[19,69939,69941],{"className":69940,"ariaHidden":62},[61],[19,69942,69944,69947],{"className":69943},[66],[19,69945],{"className":69946,"style":123},[70],[19,69948,69950,69953],{"className":69949},[75],[19,69951,49],{"className":69952},[75,76],[19,69954,69956],{"className":69955},[133],[19,69957,69959,69982],{"className":69958},[137,138],[19,69960,69962,69979],{"className":69961},[142],[19,69963,69965],{"className":69964,"style":147},[146],[19,69966,69967,69970],{"style":150},[19,69968],{"className":69969,"style":155},[154],[19,69971,69973],{"className":69972},[159,160,161,162],[19,69974,69976],{"className":69975},[75,162],[19,69977,110],{"className":69978},[75,162],[19,69980,173],{"className":69981},[172],[19,69983,69985],{"className":69984},[142],[19,69986,69988],{"className":69987,"style":180},[146],[19,69989],{},[15,69991,69992,69994,69995,67377,70038,70110,70111,54289,70234,44114,70285,3447],{},[7600,69993,10737],{}," Given that ",[19,69996,69998,70017],{"className":69997,"translate":23},[22],[19,69999,70001],{"className":70000},[27],[29,70002,70003],{"xmlns":31},[33,70004,70005,70015],{},[36,70006,70007,70009,70011,70013],{},[39,70008,41],{},[43,70010,46],{"stretchy":45},[39,70012,49],{},[43,70014,52],{"stretchy":45},[54,70016,57],{"encoding":56},[19,70018,70020],{"className":70019,"ariaHidden":62},[61],[19,70021,70023,70026,70029,70032,70035],{"className":70022},[66],[19,70024],{"className":70025,"style":71},[70],[19,70027,41],{"className":70028,"style":77},[75,76],[19,70030,46],{"className":70031},[81],[19,70033,49],{"className":70034},[75,76],[19,70036,52],{"className":70037},[88],[19,70039,70041,70058],{"className":70040,"translate":23},[22],[19,70042,70044],{"className":70043},[27],[29,70045,70046],{"xmlns":31},[33,70047,70048,70056],{},[36,70049,70050],{},[103,70051,70052,70054],{},[39,70053,49],{},[108,70055,110],{},[54,70057,113],{"encoding":56},[19,70059,70061],{"className":70060,"ariaHidden":62},[61],[19,70062,70064,70067],{"className":70063},[66],[19,70065],{"className":70066,"style":123},[70],[19,70068,70070,70073],{"className":70069},[75],[19,70071,49],{"className":70072},[75,76],[19,70074,70076],{"className":70075},[133],[19,70077,70079,70102],{"className":70078},[137,138],[19,70080,70082,70099],{"className":70081},[142],[19,70083,70085],{"className":70084,"style":147},[146],[19,70086,70087,70090],{"style":150},[19,70088],{"className":70089,"style":155},[154],[19,70091,70093],{"className":70092},[159,160,161,162],[19,70094,70096],{"className":70095},[75,162],[19,70097,110],{"className":70098},[75,162],[19,70100,173],{"className":70101},[172],[19,70103,70105],{"className":70104},[142],[19,70106,70108],{"className":70107,"style":180},[146],[19,70109],{},", by Taylor's theorem, for any ",[19,70112,70114,70146],{"className":70113,"translate":23},[22],[19,70115,70117],{"className":70116},[27],[29,70118,70119],{"xmlns":31},[33,70120,70121,70143],{},[36,70122,70123,70125,70127,70129,70131,70137,70139,70141],{},[39,70124,49],{},[43,70126,13533],{},[39,70128,13196],{},[43,70130,46],{"stretchy":45},[103,70132,70133,70135],{},[39,70134,49],{},[108,70136,110],{},[43,70138,7830],{"separator":62},[39,70140,30769],{},[43,70142,52],{"stretchy":45},[54,70144,70145],{"encoding":56},"x \\in U(x_{0},r)",[19,70147,70149,70167],{"className":70148,"ariaHidden":62},[61],[19,70150,70152,70155,70158,70161,70164],{"className":70151},[66],[19,70153],{"className":70154,"style":13562},[70],[19,70156,49],{"className":70157},[75,76],[19,70159],{"className":70160,"style":613},[411],[19,70162,13533],{"className":70163},[335],[19,70165],{"className":70166,"style":613},[411],[19,70168,70170,70173,70176,70179,70222,70225,70228,70231],{"className":70169},[66],[19,70171],{"className":70172,"style":71},[70],[19,70174,13196],{"className":70175,"style":13222},[75,76],[19,70177,46],{"className":70178},[81],[19,70180,70182,70185],{"className":70181},[75],[19,70183,49],{"className":70184},[75,76],[19,70186,70188],{"className":70187},[133],[19,70189,70191,70214],{"className":70190},[137,138],[19,70192,70194,70211],{"className":70193},[142],[19,70195,70197],{"className":70196,"style":147},[146],[19,70198,70199,70202],{"style":150},[19,70200],{"className":70201,"style":155},[154],[19,70203,70205],{"className":70204},[159,160,161,162],[19,70206,70208],{"className":70207},[75,162],[19,70209,110],{"className":70210},[75,162],[19,70212,173],{"className":70213},[172],[19,70215,70217],{"className":70216},[142],[19,70218,70220],{"className":70219,"style":180},[146],[19,70221],{},[19,70223,7830],{"className":70224},[8036],[19,70226],{"className":70227,"style":412},[411],[19,70229,30769],{"className":70230,"style":31593},[75,76],[19,70232,52],{"className":70233},[88],[19,70235,70237,70255],{"className":70236,"translate":23},[22],[19,70238,70240],{"className":70239},[27],[29,70241,70242],{"xmlns":31},[33,70243,70244,70252],{},[36,70245,70246,70248,70250],{},[39,70247,30769],{},[43,70249,42631],{},[39,70251,10758],{},[54,70253,70254],{"encoding":56},"r \\leq R",[19,70256,70258,70276],{"className":70257,"ariaHidden":62},[61],[19,70259,70261,70264,70267,70270,70273],{"className":70260},[66],[19,70262],{"className":70263,"style":45339},[70],[19,70265,30769],{"className":70266,"style":31593},[75,76],[19,70268],{"className":70269,"style":613},[411],[19,70271,42631],{"className":70272},[335],[19,70274],{"className":70275,"style":613},[411],[19,70277,70279,70282],{"className":70278},[66],[19,70280],{"className":70281,"style":7738},[70],[19,70283,10758],{"className":70284,"style":10844},[75,76],[19,70286,70288,70336],{"className":70287,"translate":23},[22],[19,70289,70291],{"className":70290},[27],[29,70292,70293],{"xmlns":31},[33,70294,70295,70333],{},[36,70296,70297,70299,70301,70303,70305,70307,70313,70315,70317,70319,70321,70327,70329,70331],{},[39,70298,41],{},[43,70300,46],{"stretchy":45},[39,70302,49],{},[43,70304,52],{"stretchy":45},[43,70306,267],{},[103,70308,70309,70311],{},[39,70310,6529],{},[39,70312,3297],{},[43,70314,46],{"stretchy":45},[39,70316,49],{},[43,70318,52],{"stretchy":45},[43,70320,2351],{},[103,70322,70323,70325],{},[39,70324,10758],{},[39,70326,3297],{},[43,70328,46],{"stretchy":45},[39,70330,49],{},[43,70332,52],{"stretchy":45},[54,70334,70335],{"encoding":56},"f(x) = T_{n}(x) + R_{n}(x)",[19,70337,70339,70366,70433],{"className":70338,"ariaHidden":62},[61],[19,70340,70342,70345,70348,70351,70354,70357,70360,70363],{"className":70341},[66],[19,70343],{"className":70344,"style":71},[70],[19,70346,41],{"className":70347,"style":77},[75,76],[19,70349,46],{"className":70350},[81],[19,70352,49],{"className":70353},[75,76],[19,70355,52],{"className":70356},[88],[19,70358],{"className":70359,"style":613},[411],[19,70361,267],{"className":70362},[335],[19,70364],{"className":70365,"style":613},[411],[19,70367,70369,70372,70415,70418,70421,70424,70427,70430],{"className":70368},[66],[19,70370],{"className":70371,"style":71},[70],[19,70373,70375,70378],{"className":70374},[75],[19,70376,6529],{"className":70377,"style":6724},[75,76],[19,70379,70381],{"className":70380},[133],[19,70382,70384,70407],{"className":70383},[137,138],[19,70385,70387,70404],{"className":70386},[142],[19,70388,70390],{"className":70389,"style":3609},[146],[19,70391,70392,70395],{"style":6739},[19,70393],{"className":70394,"style":155},[154],[19,70396,70398],{"className":70397},[159,160,161,162],[19,70399,70401],{"className":70400},[75,162],[19,70402,3297],{"className":70403},[75,76,162],[19,70405,173],{"className":70406},[172],[19,70408,70410],{"className":70409},[142],[19,70411,70413],{"className":70412,"style":180},[146],[19,70414],{},[19,70416,46],{"className":70417},[81],[19,70419,49],{"className":70420},[75,76],[19,70422,52],{"className":70423},[88],[19,70425],{"className":70426,"style":451},[411],[19,70428,2351],{"className":70429},[455],[19,70431],{"className":70432,"style":451},[411],[19,70434,70436,70439,70482,70485,70488],{"className":70435},[66],[19,70437],{"className":70438,"style":71},[70],[19,70440,70442,70445],{"className":70441},[75],[19,70443,10758],{"className":70444,"style":10844},[75,76],[19,70446,70448],{"className":70447},[133],[19,70449,70451,70474],{"className":70450},[137,138],[19,70452,70454,70471],{"className":70453},[142],[19,70455,70457],{"className":70456,"style":3609},[146],[19,70458,70459,70462],{"style":10859},[19,70460],{"className":70461,"style":155},[154],[19,70463,70465],{"className":70464},[159,160,161,162],[19,70466,70468],{"className":70467},[75,162],[19,70469,3297],{"className":70470},[75,76,162],[19,70472,173],{"className":70473},[172],[19,70475,70477],{"className":70476},[142],[19,70478,70480],{"className":70479,"style":180},[146],[19,70481],{},[19,70483,46],{"className":70484},[81],[19,70486,49],{"className":70487},[75,76],[19,70489,52],{"className":70490},[88],[15,70492,70493,70498,70499,70697,70698,44114,70749,3447],{},[7600,70494,70495,26823],{},[19,70496,70497],{},"Necessity"," If ",[19,70500,70502,70554],{"className":70501,"translate":23},[22],[19,70503,70505],{"className":70504},[27],[29,70506,70507],{"xmlns":31},[33,70508,70509,70551],{},[36,70510,70511,70513,70515,70517,70519,70521,70537],{},[39,70512,41],{},[43,70514,46],{"stretchy":45},[39,70516,49],{},[43,70518,52],{"stretchy":45},[43,70520,267],{},[103,70522,70523,70529],{},[36,70524,70525,70527],{},[39,70526,208],{},[43,70528,211],{},[36,70530,70531,70533,70535],{},[39,70532,3297],{},[43,70534,218],{},[39,70536,26855],{"mathvariant":275},[36,70538,70539,70545,70547,70549],{},[103,70540,70541,70543],{},[39,70542,6529],{},[39,70544,3297],{},[43,70546,46],{"stretchy":45},[39,70548,49],{},[43,70550,52],{"stretchy":45},[54,70552,70553],{"encoding":56},"f(x) = \\lim_{n \\rightarrow \\infty}{T_{n}(x)}",[19,70555,70557,70584],{"className":70556,"ariaHidden":62},[61],[19,70558,70560,70563,70566,70569,70572,70575,70578,70581],{"className":70559},[66],[19,70561],{"className":70562,"style":71},[70],[19,70564,41],{"className":70565,"style":77},[75,76],[19,70567,46],{"className":70568},[81],[19,70570,49],{"className":70571},[75,76],[19,70573,52],{"className":70574},[88],[19,70576],{"className":70577,"style":613},[411],[19,70579,267],{"className":70580},[335],[19,70582],{"className":70583,"style":613},[411],[19,70585,70587,70590,70639,70642],{"className":70586},[66],[19,70588],{"className":70589,"style":71},[70],[19,70591,70593,70596],{"className":70592},[304],[19,70594,208],{"className":70595},[304],[19,70597,70599],{"className":70598},[133],[19,70600,70602,70631],{"className":70601},[137,138],[19,70603,70605,70628],{"className":70604},[142],[19,70606,70608],{"className":70607,"style":3609},[146],[19,70609,70610,70613],{"style":34021},[19,70611],{"className":70612,"style":155},[154],[19,70614,70616],{"className":70615},[159,160,161,162],[19,70617,70619,70622,70625],{"className":70618},[75,162],[19,70620,3297],{"className":70621},[75,76,162],[19,70623,218],{"className":70624},[335,162],[19,70626,26855],{"className":70627},[75,162],[19,70629,173],{"className":70630},[172],[19,70632,70634],{"className":70633},[142],[19,70635,70637],{"className":70636,"style":180},[146],[19,70638],{},[19,70640],{"className":70641,"style":412},[411],[19,70643,70645,70688,70691,70694],{"className":70644},[75],[19,70646,70648,70651],{"className":70647},[75],[19,70649,6529],{"className":70650,"style":6724},[75,76],[19,70652,70654],{"className":70653},[133],[19,70655,70657,70680],{"className":70656},[137,138],[19,70658,70660,70677],{"className":70659},[142],[19,70661,70663],{"className":70662,"style":3609},[146],[19,70664,70665,70668],{"style":6739},[19,70666],{"className":70667,"style":155},[154],[19,70669,70671],{"className":70670},[159,160,161,162],[19,70672,70674],{"className":70673},[75,162],[19,70675,3297],{"className":70676},[75,76,162],[19,70678,173],{"className":70679},[172],[19,70681,70683],{"className":70682},[142],[19,70684,70686],{"className":70685,"style":180},[146],[19,70687],{},[19,70689,46],{"className":70690},[81],[19,70692,49],{"className":70693},[75,76],[19,70695,52],{"className":70696},[88],", then as ",[19,70699,70701,70719],{"className":70700,"translate":23},[22],[19,70702,70704],{"className":70703},[27],[29,70705,70706],{"xmlns":31},[33,70707,70708,70716],{},[36,70709,70710,70712,70714],{},[39,70711,3297],{},[43,70713,218],{},[39,70715,26855],{"mathvariant":275},[54,70717,70718],{"encoding":56},"n \\rightarrow \\infty",[19,70720,70722,70740],{"className":70721,"ariaHidden":62},[61],[19,70723,70725,70728,70731,70734,70737],{"className":70724},[66],[19,70726],{"className":70727,"style":3309},[70],[19,70729,3297],{"className":70730},[75,76],[19,70732],{"className":70733,"style":613},[411],[19,70735,218],{"className":70736},[335],[19,70738],{"className":70739,"style":613},[411],[19,70741,70743,70746],{"className":70742},[66],[19,70744],{"className":70745,"style":3309},[70],[19,70747,26855],{"className":70748},[75],[19,70750,70752,70797],{"className":70751,"translate":23},[22],[19,70753,70755],{"className":70754},[27],[29,70756,70757],{"xmlns":31},[33,70758,70759,70795],{},[36,70760,70761,70777,70791,70793],{},[103,70762,70763,70769],{},[36,70764,70765,70767],{},[39,70766,208],{},[43,70768,211],{},[36,70770,70771,70773,70775],{},[39,70772,3297],{},[43,70774,218],{},[39,70776,26855],{"mathvariant":275},[36,70778,70779,70785,70787,70789],{},[103,70780,70781,70783],{},[39,70782,10758],{},[39,70784,3297],{},[43,70786,46],{"stretchy":45},[39,70788,49],{},[43,70790,52],{"stretchy":45},[43,70792,267],{},[108,70794,110],{},[54,70796,69644],{"encoding":56},[19,70798,70800,70922],{"className":70799,"ariaHidden":62},[61],[19,70801,70803,70806,70855,70858,70913,70916,70919],{"className":70802},[66],[19,70804],{"className":70805,"style":71},[70],[19,70807,70809,70812],{"className":70808},[304],[19,70810,208],{"className":70811},[304],[19,70813,70815],{"className":70814},[133],[19,70816,70818,70847],{"className":70817},[137,138],[19,70819,70821,70844],{"className":70820},[142],[19,70822,70824],{"className":70823,"style":3609},[146],[19,70825,70826,70829],{"style":34021},[19,70827],{"className":70828,"style":155},[154],[19,70830,70832],{"className":70831},[159,160,161,162],[19,70833,70835,70838,70841],{"className":70834},[75,162],[19,70836,3297],{"className":70837},[75,76,162],[19,70839,218],{"className":70840},[335,162],[19,70842,26855],{"className":70843},[75,162],[19,70845,173],{"className":70846},[172],[19,70848,70850],{"className":70849},[142],[19,70851,70853],{"className":70852,"style":180},[146],[19,70854],{},[19,70856],{"className":70857,"style":412},[411],[19,70859,70861,70904,70907,70910],{"className":70860},[75],[19,70862,70864,70867],{"className":70863},[75],[19,70865,10758],{"className":70866,"style":10844},[75,76],[19,70868,70870],{"className":70869},[133],[19,70871,70873,70896],{"className":70872},[137,138],[19,70874,70876,70893],{"className":70875},[142],[19,70877,70879],{"className":70878,"style":3609},[146],[19,70880,70881,70884],{"style":10859},[19,70882],{"className":70883,"style":155},[154],[19,70885,70887],{"className":70886},[159,160,161,162],[19,70888,70890],{"className":70889},[75,162],[19,70891,3297],{"className":70892},[75,76,162],[19,70894,173],{"className":70895},[172],[19,70897,70899],{"className":70898},[142],[19,70900,70902],{"className":70901,"style":180},[146],[19,70903],{},[19,70905,46],{"className":70906},[81],[19,70908,49],{"className":70909},[75,76],[19,70911,52],{"className":70912},[88],[19,70914],{"className":70915,"style":613},[411],[19,70917,267],{"className":70918},[335],[19,70920],{"className":70921,"style":613},[411],[19,70923,70925,70928],{"className":70924},[66],[19,70926],{"className":70927,"style":1214},[70],[19,70929,110],{"className":70930},[75],[15,70932,70933,70498,70938,33437,71120,3447],{},[7600,70934,70935,26823],{},[19,70936,70937],{},"Sufficiency",[19,70939,70941,70986],{"className":70940,"translate":23},[22],[19,70942,70944],{"className":70943},[27],[29,70945,70946],{"xmlns":31},[33,70947,70948,70984],{},[36,70949,70950,70966,70980,70982],{},[103,70951,70952,70958],{},[36,70953,70954,70956],{},[39,70955,208],{},[43,70957,211],{},[36,70959,70960,70962,70964],{},[39,70961,3297],{},[43,70963,218],{},[39,70965,26855],{"mathvariant":275},[36,70967,70968,70974,70976,70978],{},[103,70969,70970,70972],{},[39,70971,10758],{},[39,70973,3297],{},[43,70975,46],{"stretchy":45},[39,70977,49],{},[43,70979,52],{"stretchy":45},[43,70981,267],{},[108,70983,110],{},[54,70985,69644],{"encoding":56},[19,70987,70989,71111],{"className":70988,"ariaHidden":62},[61],[19,70990,70992,70995,71044,71047,71102,71105,71108],{"className":70991},[66],[19,70993],{"className":70994,"style":71},[70],[19,70996,70998,71001],{"className":70997},[304],[19,70999,208],{"className":71000},[304],[19,71002,71004],{"className":71003},[133],[19,71005,71007,71036],{"className":71006},[137,138],[19,71008,71010,71033],{"className":71009},[142],[19,71011,71013],{"className":71012,"style":3609},[146],[19,71014,71015,71018],{"style":34021},[19,71016],{"className":71017,"style":155},[154],[19,71019,71021],{"className":71020},[159,160,161,162],[19,71022,71024,71027,71030],{"className":71023},[75,162],[19,71025,3297],{"className":71026},[75,76,162],[19,71028,218],{"className":71029},[335,162],[19,71031,26855],{"className":71032},[75,162],[19,71034,173],{"className":71035},[172],[19,71037,71039],{"className":71038},[142],[19,71040,71042],{"className":71041,"style":180},[146],[19,71043],{},[19,71045],{"className":71046,"style":412},[411],[19,71048,71050,71093,71096,71099],{"className":71049},[75],[19,71051,71053,71056],{"className":71052},[75],[19,71054,10758],{"className":71055,"style":10844},[75,76],[19,71057,71059],{"className":71058},[133],[19,71060,71062,71085],{"className":71061},[137,138],[19,71063,71065,71082],{"className":71064},[142],[19,71066,71068],{"className":71067,"style":3609},[146],[19,71069,71070,71073],{"style":10859},[19,71071],{"className":71072,"style":155},[154],[19,71074,71076],{"className":71075},[159,160,161,162],[19,71077,71079],{"className":71078},[75,162],[19,71080,3297],{"className":71081},[75,76,162],[19,71083,173],{"className":71084},[172],[19,71086,71088],{"className":71087},[142],[19,71089,71091],{"className":71090,"style":180},[146],[19,71092],{},[19,71094,46],{"className":71095},[81],[19,71097,49],{"className":71098},[75,76],[19,71100,52],{"className":71101},[88],[19,71103],{"className":71104,"style":613},[411],[19,71106,267],{"className":71107},[335],[19,71109],{"className":71110,"style":613},[411],[19,71112,71114,71117],{"className":71113},[66],[19,71115],{"className":71116,"style":1214},[70],[19,71118,110],{"className":71119},[75],[19,71121,71123,71174],{"className":71122,"translate":23},[22],[19,71124,71126],{"className":71125},[27],[29,71127,71128],{"xmlns":31},[33,71129,71130,71172],{},[36,71131,71132,71134,71136,71138,71140,71142,71158],{},[39,71133,41],{},[43,71135,46],{"stretchy":45},[39,71137,49],{},[43,71139,52],{"stretchy":45},[43,71141,267],{},[103,71143,71144,71150],{},[36,71145,71146,71148],{},[39,71147,208],{},[43,71149,211],{},[36,71151,71152,71154,71156],{},[39,71153,3297],{},[43,71155,218],{},[39,71157,26855],{"mathvariant":275},[36,71159,71160,71166,71168,71170],{},[103,71161,71162,71164],{},[39,71163,6529],{},[39,71165,3297],{},[43,71167,46],{"stretchy":45},[39,71169,49],{},[43,71171,52],{"stretchy":45},[54,71173,70553],{"encoding":56},[19,71175,71177,71204],{"className":71176,"ariaHidden":62},[61],[19,71178,71180,71183,71186,71189,71192,71195,71198,71201],{"className":71179},[66],[19,71181],{"className":71182,"style":71},[70],[19,71184,41],{"className":71185,"style":77},[75,76],[19,71187,46],{"className":71188},[81],[19,71190,49],{"className":71191},[75,76],[19,71193,52],{"className":71194},[88],[19,71196],{"className":71197,"style":613},[411],[19,71199,267],{"className":71200},[335],[19,71202],{"className":71203,"style":613},[411],[19,71205,71207,71210,71259,71262],{"className":71206},[66],[19,71208],{"className":71209,"style":71},[70],[19,71211,71213,71216],{"className":71212},[304],[19,71214,208],{"className":71215},[304],[19,71217,71219],{"className":71218},[133],[19,71220,71222,71251],{"className":71221},[137,138],[19,71223,71225,71248],{"className":71224},[142],[19,71226,71228],{"className":71227,"style":3609},[146],[19,71229,71230,71233],{"style":34021},[19,71231],{"className":71232,"style":155},[154],[19,71234,71236],{"className":71235},[159,160,161,162],[19,71237,71239,71242,71245],{"className":71238},[75,162],[19,71240,3297],{"className":71241},[75,76,162],[19,71243,218],{"className":71244},[335,162],[19,71246,26855],{"className":71247},[75,162],[19,71249,173],{"className":71250},[172],[19,71252,71254],{"className":71253},[142],[19,71255,71257],{"className":71256,"style":180},[146],[19,71258],{},[19,71260],{"className":71261,"style":412},[411],[19,71263,71265,71308,71311,71314],{"className":71264},[75],[19,71266,71268,71271],{"className":71267},[75],[19,71269,6529],{"className":71270,"style":6724},[75,76],[19,71272,71274],{"className":71273},[133],[19,71275,71277,71300],{"className":71276},[137,138],[19,71278,71280,71297],{"className":71279},[142],[19,71281,71283],{"className":71282,"style":3609},[146],[19,71284,71285,71288],{"style":6739},[19,71286],{"className":71287,"style":155},[154],[19,71289,71291],{"className":71290},[159,160,161,162],[19,71292,71294],{"className":71293},[75,162],[19,71295,3297],{"className":71296},[75,76,162],[19,71298,173],{"className":71299},[172],[19,71301,71303],{"className":71302},[142],[19,71304,71306],{"className":71305,"style":180},[146],[19,71307],{},[19,71309,46],{"className":71310},[81],[19,71312,49],{"className":71313},[75,76],[19,71315,52],{"className":71316},[88],[10,71318,71320],{"id":71319},"taylor-expansion","Taylor Expansion",[15,71322,71323,71324,71367,71368,71421,71422,71465,71466,71519],{},"If function ",[19,71325,71327,71346],{"className":71326,"translate":23},[22],[19,71328,71330],{"className":71329},[27],[29,71331,71332],{"xmlns":31},[33,71333,71334,71344],{},[36,71335,71336,71338,71340,71342],{},[39,71337,41],{},[43,71339,46],{"stretchy":45},[39,71341,49],{},[43,71343,52],{"stretchy":45},[54,71345,57],{"encoding":56},[19,71347,71349],{"className":71348,"ariaHidden":62},[61],[19,71350,71352,71355,71358,71361,71364],{"className":71351},[66],[19,71353],{"className":71354,"style":71},[70],[19,71356,41],{"className":71357,"style":77},[75,76],[19,71359,46],{"className":71360},[81],[19,71362,49],{"className":71363},[75,76],[19,71365,52],{"className":71366},[88]," can be expressed as the sum function of its Taylor series in a neighborhood of point ",[19,71369,71371,71388],{"className":71370,"translate":23},[22],[19,71372,71374],{"className":71373},[27],[29,71375,71376],{"xmlns":31},[33,71377,71378,71386],{},[36,71379,71380,71382,71384],{},[39,71381,49],{},[43,71383,2839],{},[108,71385,110],{},[54,71387,2844],{"encoding":56},[19,71389,71391,71409],{"className":71390,"ariaHidden":62},[61],[19,71392,71394,71397,71400,71403,71406],{"className":71393},[66],[19,71395],{"className":71396,"style":2854},[70],[19,71398,49],{"className":71399},[75,76],[19,71401],{"className":71402,"style":451},[411],[19,71404,2839],{"className":71405},[455],[19,71407],{"className":71408,"style":451},[411],[19,71410,71412,71415],{"className":71411},[66],[19,71413],{"className":71414,"style":1214},[70],[19,71416,71418],{"className":71417},[75],[19,71419,110],{"className":71420},[75],", then function ",[19,71423,71425,71444],{"className":71424,"translate":23},[22],[19,71426,71428],{"className":71427},[27],[29,71429,71430],{"xmlns":31},[33,71431,71432,71442],{},[36,71433,71434,71436,71438,71440],{},[39,71435,41],{},[43,71437,46],{"stretchy":45},[39,71439,49],{},[43,71441,52],{"stretchy":45},[54,71443,57],{"encoding":56},[19,71445,71447],{"className":71446,"ariaHidden":62},[61],[19,71448,71450,71453,71456,71459,71462],{"className":71449},[66],[19,71451],{"className":71452,"style":71},[70],[19,71454,41],{"className":71455,"style":77},[75,76],[19,71457,46],{"className":71458},[81],[19,71460,49],{"className":71461},[75,76],[19,71463,52],{"className":71464},[88]," can be expanded into a Taylor series in the neighborhood of point 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series",[19,72598,72600],{"className":72599,"translate":23},[187],[19,72601,72603,72642],{"className":72602,"translate":23},[22],[19,72604,72606],{"className":72605},[27],[29,72607,72608],{"xmlns":31,"display":196},[33,72609,72610,72640],{},[36,72611,72612,72626],{},[26840,72613,72614,72616,72624],{},[43,72615,26844],{},[36,72617,72618,72620,72622],{},[39,72619,3297],{},[43,72621,267],{},[108,72623,110],{},[39,72625,26855],{"mathvariant":275},[36,72627,72628,72634],{},[103,72629,72630,72632],{},[39,72631,3484],{},[39,72633,3297],{},[269,72635,72636,72638],{},[39,72637,49],{},[39,72639,3297],{},[54,72641,28591],{"encoding":56},[19,72643,72645],{"className":72644,"ariaHidden":62},[61],[19,72646,72648,72651,72718,72721],{"className":72647},[66],[19,72649],{"className":72650,"style":26984},[70],[19,72652,72654],{"className":72653},[304,305],[19,72655,72657,72710],{"className":72656},[137,138],[19,72658,72660,72707],{"className":72659},[142],[19,72661,72663,72683,72693],{"className":72662,"style":26997},[146],[19,72664,72665,72668],{"style":27000},[19,72666],{"className":72667,"style":27004},[154],[19,72669,72671],{"className":72670},[159,160,161,162],[19,72672,72674,72677,72680],{"className":72673},[75,162],[19,72675,3297],{"className":72676},[75,76,162],[19,72678,267],{"className":72679},[335,162],[19,72681,110],{"className":72682},[75,162],[19,72684,72685,72688],{"style":27022},[19,72686],{"className":72687,"style":27004},[154],[19,72689,72690],{},[19,72691,26844],{"className":72692},[304,27031,27032],[19,72694,72695,72698],{"style":27035},[19,72696],{"className":72697,"style":27004},[154],[19,72699,72701],{"className":72700},[159,160,161,162],[19,72702,72704],{"className":72703},[75,162],[19,72705,26855],{"className":72706},[75,162],[19,72708,173],{"className":72709},[172],[19,72711,72713],{"className":72712},[142],[19,72714,72716],{"className":72715,"style":27057},[146],[19,72717],{},[19,72719],{"className":72720,"style":412},[411],[19,72722,72724,72767],{"className":72723},[75],[19,72725,72727,72730],{"className":72726},[75],[19,72728,3484],{"className":72729},[75,76],[19,72731,72733],{"className":72732},[133],[19,72734,72736,72759],{"className":72735},[137,138],[19,72737,72739,72756],{"className":72738},[142],[19,72740,72742],{"className":72741,"style":3609},[146],[19,72743,72744,72747],{"style":150},[19,72745],{"className":72746,"style":155},[154],[19,72748,72750],{"className":72749},[159,160,161,162],[19,72751,72753],{"className":72752},[75,162],[19,72754,3297],{"className":72755},[75,76,162],[19,72757,173],{"className":72758},[172],[19,72760,72762],{"className":72761},[142],[19,72763,72765],{"className":72764,"style":180},[146],[19,72766],{},[19,72768,72770,72773],{"className":72769},[75],[19,72771,49],{"className":72772},[75,76],[19,72774,72776],{"className":72775},[133],[19,72777,72779],{"className":72778},[137],[19,72780,72782],{"className":72781},[142],[19,72783,72785],{"className":72784,"style":4148},[146],[19,72786,72787,72790],{"style":648},[19,72788],{"className":72789,"style":155},[154],[19,72791,72793],{"className":72792},[159,160,161,162],[19,72794,72796],{"className":72795},[75,162],[19,72797,3297],{"className":72798},[75,76,162],[15,72800,72801,72802,72858,72859,73024,73025,73068,73069,73119],{},"in the convergence interval ",[19,72803,72805,72828],{"className":72804,"translate":23},[22],[19,72806,72808],{"className":72807},[27],[29,72809,72810],{"xmlns":31},[33,72811,72812,72826],{},[36,72813,72814,72816,72818,72820,72822,72824],{},[43,72815,46],{"stretchy":45},[43,72817,240],{},[39,72819,10758],{},[43,72821,7830],{"separator":62},[39,72823,10758],{},[43,72825,52],{"stretchy":45},[54,72827,58453],{"encoding":56},[19,72829,72831],{"className":72830,"ariaHidden":62},[61],[19,72832,72834,72837,72840,72843,72846,72849,72852,72855],{"className":72833},[66],[19,72835],{"className":72836,"style":71},[70],[19,72838,46],{"className":72839},[81],[19,72841,240],{"className":72842},[75],[19,72844,10758],{"className":72845,"style":10844},[75,76],[19,72847,7830],{"className":72848},[8036],[19,72850],{"className":72851,"style":412},[411],[19,72853,10758],{"className":72854,"style":10844},[75,76],[19,72856,52],{"className":72857},[88],", then the coefficients of ",[19,72860,72862,72904],{"className":72861,"translate":23},[22],[19,72863,72865],{"className":72864},[27],[29,72866,72867],{"xmlns":31},[33,72868,72869,72901],{},[36,72870,72871,72873,72875,72887],{},[43,72872,26844],{},[43,72874,2839],{},[269,72876,72877,72885],{},[36,72878,72879,72881,72883],{},[39,72880,3297],{},[43,72882,267],{},[108,72884,110],{},[39,72886,26855],{"mathvariant":275},[36,72888,72889,72891,72893,72895],{},[39,72890,3484],{},[43,72892,2839],{},[39,72894,3297],{},[269,72896,72897,72899],{},[39,72898,49],{},[39,72900,3297],{},[54,72902,72903],{"encoding":56},"\\sum*{n=0}^{\\infty}{a*{n}x^{n}}",[19,72905,72907],{"className":72906,"ariaHidden":62},[61],[19,72908,72910,72913,72916,72919,72922,72971],{"className":72909},[66],[19,72911],{"className":72912,"style":71},[70],[19,72914,26844],{"className":72915,"style":31937},[304,27031,31936],[19,72917],{"className":72918,"style":412},[411],[19,72920,2839],{"className":72921},[75],[19,72923,72925,72943],{"className":72924},[75],[19,72926,72928,72931,72934,72937,72940],{"className":72927},[75],[19,72929,3297],{"className":72930},[75,76],[19,72932],{"className":72933,"style":613},[411],[19,72935,267],{"className":72936},[335],[19,72938],{"className":72939,"style":613},[411],[19,72941,110],{"className":72942},[75],[19,72944,72946],{"className":72945},[133],[19,72947,72949],{"className":72948},[137],[19,72950,72952],{"className":72951},[142],[19,72953,72956],{"className":72954,"style":72955},[146],"height:0.6987em;",[19,72957,72959,72962],{"style":72958},"top:-3.0973em;margin-right:0.05em;",[19,72960],{"className":72961,"style":155},[154],[19,72963,72965],{"className":72964},[159,160,161,162],[19,72966,72968],{"className":72967},[75,162],[19,72969,26855],{"className":72970},[75,162],[19,72972,72974,72977,72980,72983,72986,72992],{"className":72973},[75],[19,72975,3484],{"className":72976},[75,76],[19,72978],{"className":72979,"style":451},[411],[19,72981,2839],{"className":72982},[455],[19,72984],{"className":72985,"style":451},[411],[19,72987,72989],{"className":72988},[75],[19,72990,3297],{"className":72991},[75,76],[19,72993,72995,72998],{"className":72994},[75],[19,72996,49],{"className":72997},[75,76],[19,72999,73001],{"className":73000},[133],[19,73002,73004],{"className":73003},[137],[19,73005,73007],{"className":73006},[142],[19,73008,73010],{"className":73009,"style":3429},[146],[19,73011,73012,73015],{"style":1643},[19,73013],{"className":73014,"style":155},[154],[19,73016,73018],{"className":73017},[159,160,161,162],[19,73019,73021],{"className":73020},[75,162],[19,73022,3297],{"className":73023},[75,76,162]," are determined by the sum function ",[19,73026,73028,73047],{"className":73027,"translate":23},[22],[19,73029,73031],{"className":73030},[27],[29,73032,73033],{"xmlns":31},[33,73034,73035,73045],{},[36,73036,73037,73039,73041,73043],{},[39,73038,41],{},[43,73040,46],{"stretchy":45},[39,73042,49],{},[43,73044,52],{"stretchy":45},[54,73046,57],{"encoding":56},[19,73048,73050],{"className":73049,"ariaHidden":62},[61],[19,73051,73053,73056,73059,73062,73065],{"className":73052},[66],[19,73054],{"className":73055,"style":71},[70],[19,73057,41],{"className":73058,"style":77},[75,76],[19,73060,46],{"className":73061},[81],[19,73063,49],{"className":73064},[75,76],[19,73066,52],{"className":73067},[88]," and the values of its various order derivatives at ",[19,73070,73072,73089],{"className":73071,"translate":23},[22],[19,73073,73075],{"className":73074},[27],[29,73076,73077],{"xmlns":31},[33,73078,73079,73087],{},[36,73080,73081,73083,73085],{},[39,73082,49],{},[43,73084,267],{},[108,73086,110],{},[54,73088,51806],{"encoding":56},[19,73090,73092,73110],{"className":73091,"ariaHidden":62},[61],[19,73093,73095,73098,73101,73104,73107],{"className":73094},[66],[19,73096],{"className":73097,"style":3309},[70],[19,73099,49],{"className":73100},[75,76],[19,73102],{"className":73103,"style":613},[411],[19,73105,267],{"className":73106},[335],[19,73108],{"className":73109,"style":613},[411],[19,73111,73113,73116],{"className":73112},[66],[19,73114],{"className":73115,"style":1214},[70],[19,73117,110],{"className":73118},[75],", that is,",[19,73121,73123],{"className":73122,"translate":23},[187],[19,73124,73126,73212],{"className":73125,"translate":23},[22],[19,73127,73129],{"className":73128},[27],[29,73130,73131],{"xmlns":31,"display":196},[33,73132,73133,73209],{},[36,73134,73135,73141,73143,73145,73147,73149,73151,73153,73159,73161,73189,73191,73193,73195,73197,73199,73201,73203,73205,73207],{},[103,73136,73137,73139],{},[39,73138,3484],{},[108,73140,110],{},[43,73142,267],{},[39,73144,41],{},[43,73146,46],{"stretchy":45},[108,73148,110],{},[43,73150,52],{"stretchy":45},[411,73152],{"linebreak":4283},[103,73154,73155,73157],{},[39,73156,3484],{},[39,73158,3297],{},[43,73160,267],{},[226,73162,73163,73183],{},[36,73164,73165,73177,73179,73181],{},[269,73166,73167,73169],{},[39,73168,41],{},[36,73170,73171,73173,73175],{},[43,73172,46],{"stretchy":45},[39,73174,3297],{},[43,73176,52],{"stretchy":45},[43,73178,46],{"stretchy":45},[108,73180,110],{},[43,73182,52],{"stretchy":45},[36,73184,73185,73187],{},[39,73186,3297],{},[43,73188,4341],{"stretchy":45},[411,73190],{"linebreak":4283},[39,73192,3297],{},[43,73194,267],{},[108,73196,3495],{},[43,73198,7830],{"separator":62},[108,73200,3518],{},[43,73202,7830],{"separator":62},[39,73204,3447],{"mathvariant":275},[39,73206,3447],{"mathvariant":275},[39,73208,3447],{"mathvariant":275},[54,73210,73211],{"encoding":56},"a_{0}=f(0) \\\\ a_{n}=\\frac{f^{(n)}(0)}{n!} \\\\ n=1,2,...",[19,73213,73215,73273,73291,73294,73352,73467,73470,73488],{"className":73214,"ariaHidden":62},[61],[19,73216,73218,73221,73264,73267,73270],{"className":73217},[66],[19,73219],{"className":73220,"style":123},[70],[19,73222,73224,73227],{"className":73223},[75],[19,73225,3484],{"className":73226},[75,76],[19,73228,73230],{"className":73229},[133],[19,73231,73233,73256],{"className":73232},[137,138],[19,73234,73236,73253],{"className":73235},[142],[19,73237,73239],{"className":73238,"style":147},[146],[19,73240,73241,73244],{"style":150},[19,73242],{"className":73243,"style":155},[154],[19,73245,73247],{"className":73246},[159,160,161,162],[19,73248,73250],{"className":73249},[75,162],[19,73251,110],{"className":73252},[75,162],[19,73254,173],{"className":73255},[172],[19,73257,73259],{"className":73258},[142],[19,73260,73262],{"className":73261,"style":180},[146],[19,73263],{},[19,73265],{"className":73266,"style":613},[411],[19,73268,267],{"className":73269},[335],[19,73271],{"className":73272,"style":613},[411],[19,73274,73276,73279,73282,73285,73288],{"className":73275},[66],[19,73277],{"className":73278,"style":71},[70],[19,73280,41],{"className":73281,"style":77},[75,76],[19,73283,46],{"className":73284},[81],[19,73286,110],{"className":73287},[75],[19,73289,52],{"className":73290},[88],[19,73292],{"className":73293},[411,4283],[19,73295,73297,73300,73343,73346,73349],{"className":73296},[66],[19,73298],{"className":73299,"style":123},[70],[19,73301,73303,73306],{"className":73302},[75],[19,73304,3484],{"className":73305},[75,76],[19,73307,73309],{"className":73308},[133],[19,73310,73312,73335],{"className":73311},[137,138],[19,73313,73315,73332],{"className":73314},[142],[19,73316,73318],{"className":73317,"style":3609},[146],[19,73319,73320,73323],{"style":150},[19,73321],{"className":73322,"style":155},[154],[19,73324,73326],{"className":73325},[159,160,161,162],[19,73327,73329],{"className":73328},[75,162],[19,73330,3297],{"className":73331},[75,76,162],[19,73333,173],{"className":73334},[172],[19,73336,73338],{"className":73337},[142],[19,73339,73341],{"className":73340,"style":180},[146],[19,73342],{},[19,73344],{"className":73345,"style":613},[411],[19,73347,267],{"className":73348},[335],[19,73350],{"className":73351,"style":613},[411],[19,73353,73355,73358],{"className":73354},[66],[19,73356],{"className":73357,"style":7348},[70],[19,73359,73361,73364,73464],{"className":73360},[75],[19,73362],{"className":73363},[81,422],[19,73365,73367],{"className":73366},[226],[19,73368,73370,73456],{"className":73369},[137,138],[19,73371,73373,73453],{"className":73372},[142],[19,73374,73376,73390,73398],{"className":73375,"style":7367},[146],[19,73377,73378,73381],{"style":438},[19,73379],{"className":73380,"style":322},[154],[19,73382,73384,73387],{"className":73383},[75],[19,73385,3297],{"className":73386},[75,76],[19,73388,4341],{"className":73389},[88],[19,73391,73392,73395],{"style":504},[19,73393],{"className":73394,"style":322},[154],[19,73396],{"className":73397,"style":512},[511],[19,73399,73400,73403],{"style":515},[19,73401],{"className":73402,"style":322},[154],[19,73404,73406,73444,73447,73450],{"className":73405},[75],[19,73407,73409,73412],{"className":73408},[75],[19,73410,41],{"className":73411,"style":77},[75,76],[19,73413,73415],{"className":73414},[133],[19,73416,73418],{"className":73417},[137],[19,73419,73421],{"className":73420},[142],[19,73422,73424],{"className":73423,"style":7416},[146],[19,73425,73426,73429],{"style":1643},[19,73427],{"className":73428,"style":155},[154],[19,73430,73432],{"className":73431},[159,160,161,162],[19,73433,73435,73438,73441],{"className":73434},[75,162],[19,73436,46],{"className":73437},[81,162],[19,73439,3297],{"className":73440},[75,76,162],[19,73442,52],{"className":73443},[88,162],[19,73445,46],{"className":73446},[81],[19,73448,110],{"className":73449},[75],[19,73451,52],{"className":73452},[88],[19,73454,173],{"className":73455},[172],[19,73457,73459],{"className":73458},[142],[19,73460,73462],{"className":73461,"style":5671},[146],[19,73463],{},[19,73465],{"className":73466},[88,422],[19,73468],{"className":73469},[411,4283],[19,73471,73473,73476,73479,73482,73485],{"className":73472},[66],[19,73474],{"className":73475,"style":3309},[70],[19,73477,3297],{"className":73478},[75,76],[19,73480],{"className":73481,"style":613},[411],[19,73483,267],{"className":73484},[335],[19,73486],{"className":73487,"style":613},[411],[19,73489,73491,73494,73497,73500,73503,73506,73509,73512],{"className":73490},[66],[19,73492],{"className":73493,"style":8618},[70],[19,73495,3495],{"className":73496},[75],[19,73498,7830],{"className":73499},[8036],[19,73501],{"className":73502,"style":412},[411],[19,73504,3518],{"className":73505},[75],[19,73507,7830],{"className":73508},[8036],[19,73510],{"className":73511,"style":412},[411],[19,73513,4007],{"className":73514},[75],[15,73516,73517,73518,73680,73681,59161,73724,3447],{},"That is, ",[19,73519,73521,73562],{"className":73520,"translate":23},[22],[19,73522,73524],{"className":73523},[27],[29,73525,73526],{"xmlns":31},[33,73527,73528,73560],{},[36,73529,73530,73532,73534,73546],{},[43,73531,26844],{},[43,73533,2839],{},[269,73535,73536,73544],{},[36,73537,73538,73540,73542],{},[39,73539,3297],{},[43,73541,267],{},[108,73543,110],{},[39,73545,26855],{"mathvariant":275},[36,73547,73548,73550,73552,73554],{},[39,73549,3484],{},[43,73551,2839],{},[39,73553,3297],{},[269,73555,73556,73558],{},[39,73557,49],{},[39,73559,3297],{},[54,73561,72903],{"encoding":56},[19,73563,73565],{"className":73564,"ariaHidden":62},[61],[19,73566,73568,73571,73574,73577,73580,73627],{"className":73567},[66],[19,73569],{"className":73570,"style":71},[70],[19,73572,26844],{"className":73573,"style":31937},[304,27031,31936],[19,73575],{"className":73576,"style":412},[411],[19,73578,2839],{"className":73579},[75],[19,73581,73583,73601],{"className":73582},[75],[19,73584,73586,73589,73592,73595,73598],{"className":73585},[75],[19,73587,3297],{"className":73588},[75,76],[19,73590],{"className":73591,"style":613},[411],[19,73593,267],{"className":73594},[335],[19,73596],{"className":73597,"style":613},[411],[19,73599,110],{"className":73600},[75],[19,73602,73604],{"className":73603},[133],[19,73605,73607],{"className":73606},[137],[19,73608,73610],{"className":73609},[142],[19,73611,73613],{"className":73612,"style":72955},[146],[19,73614,73615,73618],{"style":72958},[19,73616],{"className":73617,"style":155},[154],[19,73619,73621],{"className":73620},[159,160,161,162],[19,73622,73624],{"className":73623},[75,162],[19,73625,26855],{"className":73626},[75,162],[19,73628,73630,73633,73636,73639,73642,73648],{"className":73629},[75],[19,73631,3484],{"className":73632},[75,76],[19,73634],{"className":73635,"style":451},[411],[19,73637,2839],{"className":73638},[455],[19,73640],{"className":73641,"style":451},[411],[19,73643,73645],{"className":73644},[75],[19,73646,3297],{"className":73647},[75,76],[19,73649,73651,73654],{"className":73650},[75],[19,73652,49],{"className":73653},[75,76],[19,73655,73657],{"className":73656},[133],[19,73658,73660],{"className":73659},[137],[19,73661,73663],{"className":73662},[142],[19,73664,73666],{"className":73665,"style":3429},[146],[19,73667,73668,73671],{"style":1643},[19,73669],{"className":73670,"style":155},[154],[19,73672,73674],{"className":73673},[159,160,161,162],[19,73675,73677],{"className":73676},[75,162],[19,73678,3297],{"className":73679},[75,76,162]," is the Taylor expansion of function ",[19,73682,73684,73703],{"className":73683,"translate":23},[22],[19,73685,73687],{"className":73686},[27],[29,73688,73689],{"xmlns":31},[33,73690,73691,73701],{},[36,73692,73693,73695,73697,73699],{},[39,73694,41],{},[43,73696,46],{"stretchy":45},[39,73698,49],{},[43,73700,52],{"stretchy":45},[54,73702,57],{"encoding":56},[19,73704,73706],{"className":73705,"ariaHidden":62},[61],[19,73707,73709,73712,73715,73718,73721],{"className":73708},[66],[19,73710],{"className":73711,"style":71},[70],[19,73713,41],{"className":73714,"style":77},[75,76],[19,73716,46],{"className":73717},[81],[19,73719,49],{"className":73720},[75,76],[19,73722,52],{"className":73723},[88],[19,73725,73727,73750],{"className":73726,"translate":23},[22],[19,73728,73730],{"className":73729},[27],[29,73731,73732],{"xmlns":31},[33,73733,73734,73748],{},[36,73735,73736,73738,73740,73742,73744,73746],{},[43,73737,46],{"stretchy":45},[43,73739,240],{},[39,73741,10758],{},[43,73743,7830],{"separator":62},[39,73745,10758],{},[43,73747,52],{"stretchy":45},[54,73749,58453],{"encoding":56},[19,73751,73753],{"className":73752,"ariaHidden":62},[61],[19,73754,73756,73759,73762,73765,73768,73771,73774,73777],{"className":73755},[66],[19,73757],{"className":73758,"style":71},[70],[19,73760,46],{"className":73761},[81],[19,73763,240],{"className":73764},[75],[19,73766,10758],{"className":73767,"style":10844},[75,76],[19,73769,7830],{"className":73770},[8036],[19,73772],{"className":73773,"style":412},[411],[19,73775,10758],{"className":73776,"style":10844},[75,76],[19,73778,52],{"className":73779},[88],[15,73781,73782],{},"The above is the process of analyzing these four mathematical concepts. In the part of Taylor series, the relevant concepts of power series are elaborated in detail. Here we summarize these four concepts:",[15,73784,73785,73788],{},[7600,73786,73787],{},"Summary:"," From the chronological point of view, we usually encounter the Taylor formula for the first time in the part of the mean value theorem of differentiation in the first volume of mathematical analysis, that is, the Taylor formula with the Peano remainder term, which is introduced by the finite increment formula in the derivative:",[19,73790,73792],{"className":73791,"translate":23},[187],[19,73793,73795,73879],{"className":73794,"translate":23},[22],[19,73796,73798],{"className":73797},[27],[29,73799,73800],{"xmlns":31,"display":196},[33,73801,73802,73876],{},[36,73803,73804,73806,73808,73810,73812,73814,73816,73818,73824,73826,73828,73834,73836,73842,73844,73846,73848,73850,73856,73858,73860,73862,73864,73866,73868,73874],{},[39,73805,41],{},[43,73807,46],{"stretchy":45},[39,73809,49],{},[43,73811,52],{"stretchy":45},[43,73813,267],{},[39,73815,41],{},[43,73817,46],{"stretchy":45},[103,73819,73820,73822],{},[39,73821,49],{},[108,73823,110],{},[43,73825,52],{"stretchy":45},[43,73827,2351],{},[269,73829,73830,73832],{},[39,73831,41],{},[43,73833,277],{"mathvariant":275,"lspace":276,"rspace":276},[43,73835,46],{"stretchy":45},[103,73837,73838,73840],{},[39,73839,49],{},[108,73841,110],{},[43,73843,52],{"stretchy":45},[43,73845,46],{"stretchy":45},[39,73847,49],{},[43,73849,240],{},[103,73851,73852,73854],{},[39,73853,49],{},[108,73855,110],{},[43,73857,52],{"stretchy":45},[43,73859,2351],{},[39,73861,1874],{},[43,73863,46],{"stretchy":45},[39,73865,49],{},[43,73867,240],{},[103,73869,73870,73872],{},[39,73871,49],{},[108,73873,110],{},[43,73875,52],{"stretchy":45},[54,73877,73878],{"encoding":56},"f(x)=f(x_{0})+f'(x_{0})(x-x_{0})+o(x-x_{0})",[19,73880,73882,73909,73976,74078,74139,74163],{"className":73881,"ariaHidden":62},[61],[19,73883,73885,73888,73891,73894,73897,73900,73903,73906],{"className":73884},[66],[19,73886],{"className":73887,"style":71},[70],[19,73889,41],{"className":73890,"style":77},[75,76],[19,73892,46],{"className":73893},[81],[19,73895,49],{"className":73896},[75,76],[19,73898,52],{"className":73899},[88],[19,73901],{"className":73902,"style":613},[411],[19,73904,267],{"className":73905},[335],[19,73907],{"className":73908,"style":613},[411],[19,73910,73912,73915,73918,73921,73964,73967,73970,73973],{"className":73911},[66],[19,73913],{"className":73914,"style":71},[70],[19,73916,41],{"className":73917,"style":77},[75,76],[19,73919,46],{"className":73920},[81],[19,73922,73924,73927],{"className":73923},[75],[19,73925,49],{"className":73926},[75,76],[19,73928,73930],{"className":73929},[133],[19,73931,73933,73956],{"className":73932},[137,138],[19,73934,73936,73953],{"className":73935},[142],[19,73937,73939],{"className":73938,"style":147},[146],[19,73940,73941,73944],{"style":150},[19,73942],{"className":73943,"style":155},[154],[19,73945,73947],{"className":73946},[159,160,161,162],[19,73948,73950],{"className":73949},[75,162],[19,73951,110],{"className":73952},[75,162],[19,73954,173],{"className":73955},[172],[19,73957,73959],{"className":73958},[142],[19,73960,73962],{"className":73961,"style":180},[146],[19,73963],{},[19,73965,52],{"className":73966},[88],[19,73968],{"className":73969,"style":451},[411],[19,73971,2351],{"className":73972},[455],[19,73974],{"className":73975,"style":451},[411],[19,73977,73979,73982,74014,74017,74060,74063,74066,74069,74072,74075],{"className":73978},[66],[19,73980],{"className":73981,"style":626},[70],[19,73983,73985,73988],{"className":73984},[75],[19,73986,41],{"className":73987,"style":77},[75,76],[19,73989,73991],{"className":73990},[133],[19,73992,73994],{"className":73993},[137],[19,73995,73997],{"className":73996},[142],[19,73998,74000],{"className":73999,"style":645},[146],[19,74001,74002,74005],{"style":648},[19,74003],{"className":74004,"style":155},[154],[19,74006,74008],{"className":74007},[159,160,161,162],[19,74009,74011],{"className":74010},[75,162],[19,74012,277],{"className":74013},[75,162],[19,74015,46],{"className":74016},[81],[19,74018,74020,74023],{"className":74019},[75],[19,74021,49],{"className":74022},[75,76],[19,74024,74026],{"className":74025},[133],[19,74027,74029,74052],{"className":74028},[137,138],[19,74030,74032,74049],{"className":74031},[142],[19,74033,74035],{"className":74034,"style":147},[146],[19,74036,74037,74040],{"style":150},[19,74038],{"className":74039,"style":155},[154],[19,74041,74043],{"className":74042},[159,160,161,162],[19,74044,74046],{"className":74045},[75,162],[19,74047,110],{"className":74048},[75,162],[19,74050,173],{"className":74051},[172],[19,74053,74055],{"className":74054},[142],[19,74056,74058],{"className":74057,"style":180},[146],[19,74059],{},[19,74061,52],{"className":74062},[88],[19,74064,46],{"className":74065},[81],[19,74067,49],{"className":74068},[75,76],[19,74070],{"className":74071,"style":451},[411],[19,74073,240],{"className":74074},[455],[19,74076],{"className":74077,"style":451},[411],[19,74079,74081,74084,74127,74130,74133,74136],{"className":74080},[66],[19,74082],{"className":74083,"style":71},[70],[19,74085,74087,74090],{"className":74086},[75],[19,74088,49],{"className":74089},[75,76],[19,74091,74093],{"className":74092},[133],[19,74094,74096,74119],{"className":74095},[137,138],[19,74097,74099,74116],{"className":74098},[142],[19,74100,74102],{"className":74101,"style":147},[146],[19,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using the finite increment formula for estimation, the error is ",[19,74219,74221,74249],{"className":74220,"translate":23},[22],[19,74222,74224],{"className":74223},[27],[29,74225,74226],{"xmlns":31},[33,74227,74228,74246],{},[36,74229,74230,74232,74234,74236,74238,74240,74242,74244],{},[39,74231,1874],{},[43,74233,46],{"stretchy":45},[39,74235,49],{},[43,74237,240],{},[39,74239,49],{},[43,74241,2839],{},[108,74243,110],{},[43,74245,52],{"stretchy":45},[54,74247,74248],{"encoding":56},"o(x-x*{0})",[19,74250,74252,74276,74294],{"className":74251,"ariaHidden":62},[61],[19,74253,74255,74258,74261,74264,74267,74270,74273],{"className":74254},[66],[19,74256],{"className":74257,"style":71},[70],[19,74259,1874],{"className":74260},[75,76],[19,74262,46],{"className":74263},[81],[19,74265,49],{"className":74266},[75,76],[19,74268],{"className":74269,"style":451},[411],[19,74271,240],{"className":74272},[455],[19,74274],{"className":74275,"style":451},[411],[19,74277,74279,74282,74285,74288,74291],{"className":74278},[66],[19,74280],{"className":74281,"style":2854},[70],[19,74283,49],{"className":74284},[75,76],[19,74286],{"className":74287,"style":451},[411],[19,74289,2839],{"className":74290},[455],[19,74292],{"className":74293,"style":451},[411],[19,74295,74297,74300,74306],{"className":74296},[66],[19,74298],{"className":74299,"style":71},[70],[19,74301,74303],{"className":74302},[75],[19,74304,110],{"className":74305},[75],[19,74307,52],{"className":74308},[88],", and its estimation effect is quite rough. If the function is known to have an ",[19,74311,74313,74326],{"className":74312,"translate":23},[22],[19,74314,74316],{"className":74315},[27],[29,74317,74318],{"xmlns":31},[33,74319,74320,74324],{},[36,74321,74322],{},[39,74323,3297],{},[54,74325,3297],{"encoding":56},[19,74327,74329],{"className":74328,"ariaHidden":62},[61],[19,74330,74332,74335],{"className":74331},[66],[19,74333],{"className":74334,"style":3309},[70],[19,74336,3297],{"className":74337},[75,76],"-th order derivative at the point ",[19,74340,74342,74359],{"className":74341,"translate":23},[22],[19,74343,74345],{"className":74344},[27],[29,74346,74347],{"xmlns":31},[33,74348,74349,74357],{},[36,74350,74351,74353,74355],{},[39,74352,49],{},[43,74354,2839],{},[108,74356,110],{},[54,74358,2844],{"encoding":56},[19,74360,74362,74380],{"className":74361,"ariaHidden":62},[61],[19,74363,74365,74368,74371,74374,74377],{"className":74364},[66],[19,74366],{"className":74367,"style":2854},[70],[19,74369,49],{"className":74370},[75,76],[19,74372],{"className":74373,"style":451},[411],[19,74375,2839],{"className":74376},[455],[19,74378],{"className":74379,"style":451},[411],[19,74381,74383,74386],{"className":74382},[66],[19,74384],{"className":74385,"style":1214},[70],[19,74387,74389],{"className":74388},[75],[19,74390,110],{"className":74391},[75],", we can generalize the finite increment formula, that is, to obtain the Taylor formula with the Peano remainder term. Its estimation error is ",[19,74394,74396,74431],{"className":74395,"translate":23},[22],[19,74397,74399],{"className":74398},[27],[29,74400,74401],{"xmlns":31},[33,74402,74403,74429],{},[36,74404,74405,74407,74409,74411,74413,74415,74417,74419,74421,74427],{},[39,74406,1874],{},[43,74408,46],{"stretchy":45},[43,74410,46],{"stretchy":45},[39,74412,49],{},[43,74414,240],{},[39,74416,49],{},[43,74418,2839],{},[108,74420,110],{},[269,74422,74423,74425],{},[43,74424,52],{"stretchy":45},[39,74426,3297],{},[43,74428,52],{"stretchy":45},[54,74430,3352],{"encoding":56},[19,74432,74434,74458,74476],{"className":74433,"ariaHidden":62},[61],[19,74435,74437,74440,74443,74446,74449,74452,74455],{"className":74436},[66],[19,74438],{"className":74439,"style":71},[70],[19,74441,1874],{"className":74442},[75,76],[19,74444,3368],{"className":74445},[81],[19,74447,49],{"className":74448},[75,76],[19,74450],{"className":74451,"style":451},[411],[19,74453,240],{"className":74454},[455],[19,74456],{"className":74457,"style":451},[411],[19,74459,74461,74464,74467,74470,74473],{"className":74460},[66],[19,74462],{"className":74463,"style":2854},[70],[19,74465,49],{"className":74466},[75,76],[19,74468],{"className":74469,"style":451},[411],[19,74471,2839],{"className":74472},[455],[19,74474],{"className":74475,"style":451},[411],[19,74477,74479,74482,74488,74520],{"className":74478},[66],[19,74480],{"className":74481,"style":71},[70],[19,74483,74485],{"className":74484},[75],[19,74486,110],{"className":74487},[75],[19,74489,74491,74494],{"className":74490},[88],[19,74492,52],{"className":74493},[88],[19,74495,74497],{"className":74496},[133],[19,74498,74500],{"className":74499},[137],[19,74501,74503],{"className":74502},[142],[19,74504,74506],{"className":74505,"style":3429},[146],[19,74507,74508,74511],{"style":1643},[19,74509],{"className":74510,"style":155},[154],[19,74512,74514],{"className":74513},[159,160,161,162],[19,74515,74517],{"className":74516},[75,162],[19,74518,3297],{"className":74519},[75,76,162],[19,74521,52],{"className":74522},[88],". In order to further improve the estimation effect and quantitatively analyze the range of errors, the Taylor formula with the Lagrange remainder term is introduced. The use condition of Taylor formula with the Lagrange remainder term is further strict, which requires that the function ",[19,74525,74527,74546],{"className":74526,"translate":23},[22],[19,74528,74530],{"className":74529},[27],[29,74531,74532],{"xmlns":31},[33,74533,74534,74544],{},[36,74535,74536,74538,74540,74542],{},[39,74537,41],{},[43,74539,46],{"stretchy":45},[39,74541,49],{},[43,74543,52],{"stretchy":45},[54,74545,57],{"encoding":56},[19,74547,74549],{"className":74548,"ariaHidden":62},[61],[19,74550,74552,74555,74558,74561,74564],{"className":74551},[66],[19,74553],{"className":74554,"style":71},[70],[19,74556,41],{"className":74557,"style":77},[75,76],[19,74559,46],{"className":74560},[81],[19,74562,49],{"className":74563},[75,76],[19,74565,52],{"className":74566},[88],[19,74568,74570,74591],{"className":74569,"translate":23},[22],[19,74571,74573],{"className":74572},[27],[29,74574,74575],{"xmlns":31},[33,74576,74577,74589],{},[36,74578,74579,74581,74583,74585,74587],{},[43,74580,46],{"stretchy":45},[39,74582,3297],{},[43,74584,2351],{},[108,74586,3495],{},[43,74588,52],{"stretchy":45},[54,74590,17281],{"encoding":56},[19,74592,74594,74615],{"className":74593,"ariaHidden":62},[61],[19,74595,74597,74600,74603,74606,74609,74612],{"className":74596},[66],[19,74598],{"className":74599,"style":71},[70],[19,74601,46],{"className":74602},[81],[19,74604,3297],{"className":74605},[75,76],[19,74607],{"className":74608,"style":451},[411],[19,74610,2351],{"className":74611},[455],[19,74613],{"className":74614,"style":451},[411],[19,74616,74618,74621,74624],{"className":74617},[66],[19,74619],{"className":74620,"style":71},[70],[19,74622,3495],{"className":74623},[75],[19,74625,52],{"className":74626},[88],"-th order at the point ",[19,74629,74631,74648],{"className":74630,"translate":23},[22],[19,74632,74634],{"className":74633},[27],[29,74635,74636],{"xmlns":31},[33,74637,74638,74646],{},[36,74639,74640,74642,74644],{},[39,74641,49],{},[43,74643,2839],{},[108,74645,110],{},[54,74647,2844],{"encoding":56},[19,74649,74651,74669],{"className":74650,"ariaHidden":62},[61],[19,74652,74654,74657,74660,74663,74666],{"className":74653},[66],[19,74655],{"className":74656,"style":2854},[70],[19,74658,49],{"className":74659},[75,76],[19,74661],{"className":74662,"style":451},[411],[19,74664,2839],{"className":74665},[455],[19,74667],{"className":74668,"style":451},[411],[19,74670,74672,74675],{"className":74671},[66],[19,74673],{"className":74674,"style":1214},[70],[19,74676,74678],{"className":74677},[75],[19,74679,110],{"className":74680},[75],". Its corresponding error is",[19,74683,74685],{"className":74684,"translate":23},[187],[19,74686,74688,74764],{"className":74687,"translate":23},[22],[19,74689,74691],{"className":74690},[27],[29,74692,74693],{"xmlns":31,"display":196},[33,74694,74695,74761],{},[36,74696,74697,74737,74739,74741,74743,74749],{},[226,74698,74699,74723],{},[36,74700,74701,74717,74719,74721],{},[269,74702,74703,74705],{},[39,74704,41],{},[36,74706,74707,74709,74711,74713,74715],{},[43,74708,46],{"stretchy":45},[39,74710,3297],{},[43,74712,2351],{},[108,74714,3495],{},[43,74716,52],{"stretchy":45},[43,74718,46],{"stretchy":45},[39,74720,17519],{},[43,74722,52],{"stretchy":45},[36,74724,74725,74727,74729,74731,74733,74735],{},[43,74726,46],{"stretchy":45},[39,74728,3297],{},[43,74730,2351],{},[108,74732,3495],{},[43,74734,52],{"stretchy":45},[43,74736,4341],{"stretchy":45},[43,74738,46],{"stretchy":45},[39,74740,49],{},[43,74742,240],{},[103,74744,74745,74747],{},[39,74746,49],{},[108,74748,110],{},[269,74750,74751,74753],{},[43,74752,52],{"stretchy":45},[36,74754,74755,74757,74759],{},[39,74756,3297],{},[43,74758,2351],{},[108,74760,3495],{},[54,74762,74763],{"encoding":56},"\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-x_{0})^{n+1}",[19,74765,74767,74918],{"className":74766,"ariaHidden":62},[61],[19,74768,74770,74773,74903,74906,74909,74912,74915],{"className":74769},[66],[19,74771],{"className":74772,"style":15256},[70],[19,74774,74776,74779,74900],{"className":74775},[75],[19,74777],{"className":74778},[81,422],[19,74780,74782],{"className":74781},[226],[19,74783,74785,74892],{"className":74784},[137,138],[19,74786,74788,74889],{"className":74787},[142],[19,74789,74791,74820,74828],{"className":74790,"style":7367},[146],[19,74792,74793,74796],{"style":438},[19,74794],{"className":74795,"style":322},[154],[19,74797,74799,74802,74805,74808,74811,74814,74817],{"className":74798},[75],[19,74800,46],{"className":74801},[81],[19,74803,3297],{"className":74804},[75,76],[19,74806],{"className":74807,"style":451},[411],[19,74809,2351],{"className":74810},[455],[19,74812],{"className":74813,"style":451},[411],[19,74815,3495],{"className":74816},[75],[19,74818,18672],{"className":74819},[88],[19,74821,74822,74825],{"style":504},[19,74823],{"className":74824,"style":322},[154],[19,74826],{"className":74827,"style":512},[511],[19,74829,74830,74833],{"style":515},[19,74831],{"className":74832,"style":322},[154],[19,74834,74836,74880,74883,74886],{"className":74835},[75],[19,74837,74839,74842],{"className":74838},[75],[19,74840,41],{"className":74841,"style":77},[75,76],[19,74843,74845],{"className":74844},[133],[19,74846,74848],{"className":74847},[137],[19,74849,74851],{"className":74850},[142],[19,74852,74854],{"className":74853,"style":7416},[146],[19,74855,74856,74859],{"style":1643},[19,74857],{"className":74858,"style":155},[154],[19,74860,74862],{"className":74861},[159,160,161,162],[19,74863,74865,74868,74871,74874,74877],{"className":74864},[75,162],[19,74866,46],{"className":74867},[81,162],[19,74869,3297],{"className":74870},[75,76,162],[19,74872,2351],{"className":74873},[455,162],[19,74875,3495],{"className":74876},[75,162],[19,74878,52],{"className":74879},[88,162],[19,74881,46],{"className":74882},[81],[19,74884,17519],{"className":74885,"style":17547},[75,76],[19,74887,52],{"className":74888},[88],[19,74890,173],{"className":74891},[172],[19,74893,74895],{"className":74894},[142],[19,74896,74898],{"className":74897,"style":11497},[146],[19,74899],{},[19,74901],{"className":74902},[88,422],[19,74904,46],{"className":74905},[81],[19,74907,49],{"className":74908},[75,76],[19,74910],{"className":74911,"style":451},[411],[19,74913,240],{"className":74914},[455],[19,74916],{"className":74917,"style":451},[411],[19,74919,74921,74924,74967],{"className":74920},[66],[19,74922],{"className":74923,"style":3908},[70],[19,74925,74927,74930],{"className":74926},[75],[19,74928,49],{"className":74929},[75,76],[19,74931,74933],{"className":74932},[133],[19,74934,74936,74959],{"className":74935},[137,138],[19,74937,74939,74956],{"className":74938},[142],[19,74940,74942],{"className":74941,"style":147},[146],[19,74943,74944,74947],{"style":150},[19,74945],{"className":74946,"style":155},[154],[19,74948,74950],{"className":74949},[159,160,161,162],[19,74951,74953],{"className":74952},[75,162],[19,74954,110],{"className":74955},[75,162],[19,74957,173],{"className":74958},[172],[19,74960,74962],{"className":74961},[142],[19,74963,74965],{"className":74964,"style":180},[146],[19,74966],{},[19,74968,74970,74973],{"className":74969},[88],[19,74971,52],{"className":74972},[88],[19,74974,74976],{"className":74975},[133],[19,74977,74979],{"className":74978},[137],[19,74980,74982],{"className":74981},[142],[19,74983,74985],{"className":74984,"style":3970},[146],[19,74986,74987,74990],{"style":648},[19,74988],{"className":74989,"style":155},[154],[19,74991,74993],{"className":74992},[159,160,161,162],[19,74994,74996,74999,75002],{"className":74995},[75,162],[19,74997,3297],{"className":749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Lagrange remainder term can quantitatively analyze the error, and its estimation effect is much better than that of the Peano remainder term.",[15,75009,75010,75011,75054,75055,75127],{},"Following the learning order of the textbook, after learning related knowledge such as series, functional series, and power series in the second volume of mathematical analysis, we can introduce the concept of Taylor series. If the function ",[19,75012,75014,75033],{"className":75013,"translate":23},[22],[19,75015,75017],{"className":75016},[27],[29,75018,75019],{"xmlns":31},[33,75020,75021,75031],{},[36,75022,75023,75025,75027,75029],{},[39,75024,41],{},[43,75026,46],{"stretchy":45},[39,75028,49],{},[43,75030,52],{"stretchy":45},[54,75032,57],{"encoding":56},[19,75034,75036],{"className":75035,"ariaHidden":62},[61],[19,75037,75039,75042,75045,75048,75051],{"className":75038},[66],[19,75040],{"className":75041,"style":71},[70],[19,75043,41],{"className":75044,"style":77},[75,76],[19,75046,46],{"className":75047},[81],[19,75049,49],{"className":75050},[75,76],[19,75052,52],{"className":75053},[88]," has derivatives of any order at some point 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then we can 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expression is called the Taylor series of function 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As a special power series, the Taylor series has a convergence range, namely the convergence domain of the Taylor series, and we give the necessary and sufficient conditions for the Taylor series of function 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in a certain neighborhood ",[19,76199,76201,76229],{"className":76200,"translate":23},[22],[19,76202,76204],{"className":76203},[27],[29,76205,76206],{"xmlns":31},[33,76207,76208,76226],{},[36,76209,76210,76212,76214,76216,76218,76220,76222,76224],{},[39,76211,13196],{},[43,76213,46],{"stretchy":45},[39,76215,49],{},[43,76217,2839],{},[108,76219,110],{},[43,76221,7830],{"separator":62},[39,76223,30769],{},[43,76225,52],{"stretchy":45},[54,76227,76228],{"encoding":56},"U(x*{0},r)",[19,76230,76232,76256],{"className":76231,"ariaHidden":62},[61],[19,76233,76235,76238,76241,76244,76247,76250,76253],{"className":76234},[66],[19,76236],{"className":76237,"style":71},[70],[19,76239,13196],{"className":76240,"style":13222},[75,76],[19,76242,46],{"className":76243},[81],[19,76245,49],{"className":76246},[75,76],[19,76248],{"className":76249,"style":451},[411],[19,76251,2839],{"className":76252},[455],[19,76254],{"className":76255,"style":451},[411],[19,76257,76259,76262,76268,76271,76274,76277],{"className":76258},[66],[19,76260],{"className":76261,"style":71},[70],[19,76263,76265],{"className":76264},[75],[19,76266,110],{"className":76267},[75],[19,76269,7830],{"className":76270},[8036],[19,76272],{"className":76273,"style":412},[411],[19,76275,30769],{"className":76276,"style":31593},[75,76],[19,76278,52],{"className":76279},[88]," to be equal to its sum function: ",[19,76282,76284,76328],{"className":76283,"translate":23},[22],[19,76285,76287],{"className":76286},[27],[29,76288,76289],{"xmlns":31},[33,76290,76291,76325],{},[36,76292,76293,76295,76297,76299,76307,76321,76323],{},[39,76294,208],{},[43,76296,211],{},[43,76298,2839],{},[36,76300,76301,76303,76305],{},[39,76302,3297],{},[43,76304,218],{},[39,76306,26855],{"mathvariant":275},[36,76308,76309,76315,76317,76319],{},[103,76310,76311,76313],{},[39,76312,10758],{},[39,76314,3297],{},[43,76316,46],{"stretchy":45},[39,76318,49],{},[43,76320,52],{"stretchy":45},[43,76322,267],{},[108,76324,110],{},[54,76326,76327],{"encoding":56},"\\lim*{n \\rightarrow \\infty}{R_{n}(x)}=0",[19,76329,76331,76428],{"className":76330,"ariaHidden":62},[61],[19,76332,76334,76337,76340,76343,76346,76364,76419,76422,76425],{"className":76333},[66],[19,76335],{"className":76336,"style":71},[70],[19,76338,208],{"className":76339},[304],[19,76341],{"className":76342,"style":412},[411],[19,76344,2839],{"className":76345},[75],[19,76347,76349,76352,76355,76358,76361],{"className":76348},[75],[19,76350,3297],{"className":76351},[75,76],[19,76353],{"className":76354,"style":613},[411],[19,76356,218],{"className":76357},[335],[19,76359],{"className":76360,"style":613},[411],[19,76362,26855],{"className":76363},[75],[19,76365,76367,76410,76413,76416],{"className":76366},[75],[19,76368,76370,76373],{"className":76369},[75],[19,76371,10758],{"className":76372,"style":10844},[75,76],[19,76374,76376],{"className":76375},[133],[19,76377,76379,76402],{"className":76378},[137,138],[19,76380,76382,76399],{"className":76381},[142],[19,76383,76385],{"className":76384,"style":3609},[146],[19,76386,76387,76390],{"style":10859},[19,76388],{"className":76389,"style":155},[154],[19,76391,76393],{"className":76392},[159,160,161,162],[19,76394,76396],{"className":76395},[75,162],[19,76397,3297],{"className":76398},[75,76,162],[19,76400,173],{"className":76401},[172],[19,76403,76405],{"className":76404},[142],[19,76406,76408],{"className":76407,"style":180},[146],[19,76409],{},[19,76411,46],{"className":76412},[81],[19,76414,49],{"className":76415},[75,76],[19,76417,52],{"className":76418},[88],[19,76420],{"className":76421,"style":613},[411],[19,76423,267],{"className":76424},[335],[19,76426],{"className":76427,"style":613},[411],[19,76429,76431,76434],{"className":76430},[66],[19,76432],{"className":76433,"style":1214},[70],[19,76435,110],{"className":76436},[75],[15,76438,76439,76440,76483,76484,76527,76528,76571],{},"This leads to the concept of Taylor expansion. The so-called Taylor expansion refers to the power series expression that function ",[19,76441,76443,76462],{"className":76442,"translate":23},[22],[19,76444,76446],{"className":76445},[27],[29,76447,76448],{"xmlns":31},[33,76449,76450,76460],{},[36,76451,76452,76454,76456,76458],{},[39,76453,41],{},[43,76455,46],{"stretchy":45},[39,76457,49],{},[43,76459,52],{"stretchy":45},[54,76461,57],{"encoding":56},[19,76463,76465],{"className":76464,"ariaHidden":62},[61],[19,76466,76468,76471,76474,76477,76480],{"className":76467},[66],[19,76469],{"className":76470,"style":71},[70],[19,76472,41],{"className":76473,"style":77},[75,76],[19,76475,46],{"className":76476},[81],[19,76478,49],{"className":76479},[75,76],[19,76481,52],{"className":76482},[88]," can be expanded into in a certain neighborhood. In this neighborhood, function ",[19,76485,76487,76506],{"className":76486,"translate":23},[22],[19,76488,76490],{"className":76489},[27],[29,76491,76492],{"xmlns":31},[33,76493,76494,76504],{},[36,76495,76496,76498,76500,76502],{},[39,76497,41],{},[43,76499,46],{"stretchy":45},[39,76501,49],{},[43,76503,52],{"stretchy":45},[54,76505,57],{"encoding":56},[19,76507,76509],{"className":76508,"ariaHidden":62},[61],[19,76510,76512,76515,76518,76521,76524],{"className":76511},[66],[19,76513],{"className":76514,"style":71},[70],[19,76516,41],{"className":76517,"style":77},[75,76],[19,76519,46],{"className":76520},[81],[19,76522,49],{"className":76523},[75,76],[19,76525,52],{"className":76526},[88]," is completely characterized by the power series. This is different from the approximation by polynomial functions above, but is equal in a real sense. However, the application of power series has certain limitations, namely, because the convergence domain of power series is often a finite interval, it can only characterize function ",[19,76529,76531,76550],{"className":76530,"translate":23},[22],[19,76532,76534],{"className":76533},[27],[29,76535,76536],{"xmlns":31},[33,76537,76538,76548],{},[36,76539,76540,76542,76544,76546],{},[39,76541,41],{},[43,76543,46],{"stretchy":45},[39,76545,49],{},[43,76547,52],{"stretchy":45},[54,76549,57],{"encoding":56},[19,76551,76553],{"className":76552,"ariaHidden":62},[61],[19,76554,76556,76559,76562,76565,76568],{"className":76555},[66],[19,76557],{"className":76558,"style":71},[70],[19,76560,41],{"className":76561,"style":77},[75,76],[19,76563,46],{"className":76564},[81],[19,76566,49],{"className":76567},[75,76],[19,76569,52],{"className":76570},[88]," in a small segment interval. In order to solve this drawback of power series, research on Fourier series has emerged.",{"title":76573,"searchDepth":76574,"depth":76574,"links":76575},"",2,[76576,76577,76578,76579],{"id":12,"depth":76574,"text":13},{"id":17032,"depth":76574,"text":17033},{"id":26619,"depth":76574,"text":26620},{"id":71319,"depth":76574,"text":71320},false,true,"md",null,{},"2024-03-06","\u002Fblog\u002F2024\u002F2024-03-06-taylor-formula-taylor-theorem-taylor-series-taylor-expansion",{"title":5,"description":76573},"blog\u002F2024\u002F2024-03-06-taylor-formula-taylor-theorem-taylor-series-taylor-expansion","This article mainly introduces the content and connection between the four concepts of Taylor's formula, Taylor's theorem, Taylor series and Taylor's expansion.",[76591],"mathematics","haLw9mbZiu4IStME3psqgdvxdR6fA0zTeVnAZ5StKns",1777090297694]