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can derive the following 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we 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\\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{(2k)!!}{(2k+1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{2k}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\frac{1}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} = 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we obtain ",[4202,14556,13],{},[19,14558,14560],{"className":14559,"translate":23},[22],[19,14561,14563,14653],{"className":14562,"translate":23},[27],[19,14564,14566],{"className":14565},[31],[33,14567,14568],{"xmlns":35,"display":36},[38,14569,14570,14650],{},[41,14571,14572,14588,14642,14644],{},[10327,14573,14574,14580],{},[41,14575,14576,14578],{},[58,14577,10333],{},[47,14579,74],{},[41,14581,14582,14584,14586],{},[58,14583,4255],{},[47,14585,10342],{},[58,14587,10346],{"mathvariant":10345},[55,14589,14590,14608],{},[41,14591,14592,14594,14596,14598,14600,14602],{},[47,14593,412],{"stretchy":411},[51,14595,63],{},[58,14597,4255],{},[47,14599,422],{"stretchy":411},[47,14601,7425],{"stretchy":411},[65,14603,14604,14606],{},[47,14605,7425],{"stretchy":411},[51,14607,63],{},[41,14609,14610,14612,14614,14616,14618,14620,14622,14624,14626,14628,14630,14632,14634,14636,14638,14640],{},[47,14611,412],{"stretchy":411},[51,14613,63],{},[58,14615,4255],{},[47,14617,1066],{},[51,14619,402],{},[47,14621,422],{"stretchy":411},[47,14623,7425],{"stretchy":411},[47,14625,7425],{"stretchy":411},[47,14627,412],{"stretchy":411},[51,14629,63],{},[58,14631,4255],{},[47,14633,373],{},[51,14635,402],{},[47,14637,422],{"stretchy":411},[47,14639,7425],{"stretchy":411},[47,14641,7425],{"stretchy":411},[47,14643,370],{},[55,14645,14646,14648],{},[58,14647,60],{},[51,14649,63],{},[91,14651,14652],{"encoding":93},"\\lim_{k \\rightarrow \\infty} \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \\frac{\\pi}{2}",[19,14654,14656,14872],{"className":14655,"ariaHidden":99},[98],[19,14657,14659,14662,14715,14718,14863,14866,14869],{"className":14658},[103],[19,14660],{"className":14661,"style":12616},[107],[19,14663,14665],{"className":14664},[112,10645],[19,14666,14668,14707],{"className":14667},[126,127],[19,14669,14671,14704],{"className":14670},[131],[19,14672,14674,14694],{"className":14673,"style":4286},[135],[19,14675,14676,14679],{"style":10657},[19,14677],{"className":14678,"style":144},[143],[19,14680,14682],{"className":14681},[148,149,150,151],[19,14683,14685,14688,14691],{"className":14684},[155,151],[19,14686,4255],{"className":14687,"style":4293},[155,236,151],[19,14689,10342],{"className":14690},[611,151],[19,14692,10346],{"className":14693},[155,151],[19,14695,14696,14699],{"style":10678},[19,14697],{"className":14698,"style":144},[143],[19,14700,14701],{},[19,14702,10333],{"className":14703},[112],[19,14705,242],{"className":14706},[241],[19,14708,14710],{"className":14709},[131],[19,14711,14713],{"className":14712,"style":10696},[135],[19,14714],{},[19,14716],{"className":14717,"style":272},[271],[19,14719,14721,14724,14860],{"className":14720},[155],[19,14722],{"className":14723},[177,178],[19,14725,14727],{"className":14726},[55],[19,14728,14730,14852],{"className":14729},[126,127],[19,14731,14733,14849],{"className":14732},[131],[19,14734,14736,14792,14800],{"className":14735,"style":12635},[135],[19,14737,14738,14741],{"style":3956},[19,14739],{"className":14740,"style":144},[143],[19,14742,14744,14747,14750,14753,14756,14759,14762,14765,14768,14771,14774,14777,14780,14783,14786,14789],{"className":14743},[155],[19,14745,412],{"className":14746},[177],[19,14748,63],{"className":14749},[155],[19,14751,4255],{"className":14752,"style":4293},[155,236],[19,14754],{"className":14755,"style":920},[271],[19,14757,1066],{"className":14758},[788],[19,14760],{"className":14761,"style":920},[271],[19,14763,402],{"className":14764},[155],[19,14766,7889],{"className":14767},[255],[19,14769,412],{"className":14770},[177],[19,14772,63],{"className":14773},[155],[19,14775,4255],{"className":14776,"style":4293},[155,236],[19,14778],{"className":14779,"style":920},[271],[19,14781,373],{"className":14782},[788],[19,14784],{"className":14785,"style":920},[271],[19,14787,402],{"className":14788},[155],[19,14790,7889],{"className":14791},[255],[19,14793,14794,14797],{"style":3968},[19,14795],{"className":14796,"style":144},[143],[19,14798],{"className":14799,"style":3975},[219],[19,14801,14802,14805],{"style":3978},[19,14803],{"className":14804,"style":144},[143],[19,14806,14808,14811,14814,14817,14820],{"className":14807},[155],[19,14809,412],{"className":14810},[177],[19,14812,63],{"className":14813},[155],[19,14815,4255],{"className":14816,"style":4293},[155,236],[19,14818,12720],{"className":14819},[255],[19,14821,14823,14826],{"className":14822},[255],[19,14824,7425],{"className":14825},[255],[19,14827,14829],{"className":14828},[122],[19,14830,14832],{"className":14831},[126],[19,14833,14835],{"className":14834},[131],[19,14836,14838],{"className":14837,"style":12739},[135],[19,14839,14840,14843],{"style":12742},[19,14841],{"className":14842,"style":298},[143],[19,14844,14846],{"className":14845},[148,149,150,151],[19,14847,63],{"className":14848},[155,151],[19,14850,242],{"className":14851},[241],[19,14853,14855],{"className":14854},[131],[19,14856,14858],{"className":14857,"style":7939},[135],[19,14859],{},[19,14861],{"className":14862},[255,178],[19,14864],{"className":14865,"style":607},[271],[19,14867,370],{"className":14868},[611],[19,14870],{"className":14871,"style":607},[271],[19,14873,14875,14878],{"className":14874},[103],[19,14876],{"className":14877,"style":5766},[107],[19,14879,14881,14884,14937],{"className":14880},[155],[19,14882],{"className":14883},[177,178],[19,14885,14887],{"className":14886},[55],[19,14888,14890,14929],{"className":14889},[126,127],[19,14891,14893,14926],{"className":14892},[131],[19,14894,14896,14907,14915],{"className":14895,"style":5785},[135],[19,14897,14898,14901],{"style":3956},[19,14899],{"className":14900,"style":144},[143],[19,14902,14904],{"className":14903},[155],[19,14905,63],{"className":14906},[155],[19,14908,14909,14912],{"style":3968},[19,14910],{"className":14911,"style":144},[143],[19,14913],{"className":14914,"style":3975},[219],[19,14916,14917,14920],{"style":3978},[19,14918],{"className":14919,"style":144},[143],[19,14921,14923],{"className":14922},[155],[19,14924,60],{"className":14925,"style":237},[155,236],[19,14927,242],{"className":14928},[241],[19,14930,14932],{"className":14931},[131],[19,14933,14935],{"className":14934,"style":4009},[135],[19,14936],{},[19,14938],{"className":14939},[255,178],[15,14941,14942],{},[4202,14943,14944],{},"This formula introduces Wallis's result in preparation for deriving Stirling's formula.",[10,14946,14948],{"id":14947},"stirlings-formula","Stirling's Formula",[15,14950,14951,14952,14986,14987,15047,15048,15108],{},"Stirling's formula reveals a relationship between ",[19,14953,14955,14971],{"className":14954,"translate":23},[27],[19,14956,14958],{"className":14957},[31],[33,14959,14960],{"xmlns":35},[38,14961,14962,14968],{},[41,14963,14964,14966],{},[58,14965,77],{},[47,14967,7425],{"stretchy":411},[91,14969,14970],{"encoding":93},"n!",[19,14972,14974],{"className":14973,"ariaHidden":99},[98],[19,14975,14977,14980,14983],{"className":14976},[103],[19,14978],{"className":14979,"style":4286},[107],[19,14981,77],{"className":14982},[155,236],[19,14984,7425],{"className":14985},[255],", ",[19,14988,14990,15008],{"className":14989,"translate":23},[27],[19,14991,14993],{"className":14992},[31],[33,14994,14995],{"xmlns":35},[38,14996,14997,15005],{},[41,14998,14999],{},[65,15000,15001,15003],{},[58,15002,77],{},[58,15004,77],{},[91,15006,15007],{"encoding":93},"n^n",[19,15009,15011],{"className":15010,"ariaHidden":99},[98],[19,15012,15014,15018],{"className":15013},[103],[19,15015],{"className":15016,"style":15017},[107],"height:0.6644em;",[19,15019,15021,15024],{"className":15020},[155],[19,15022,77],{"className":15023},[155,236],[19,15025,15027],{"className":15026},[122],[19,15028,15030],{"className":15029},[126],[19,15031,15033],{"className":15032},[131],[19,15034,15036],{"className":15035,"style":15017},[135],[19,15037,15038,15041],{"style":12742},[19,15039],{"className":15040,"style":298},[143],[19,15042,15044],{"className":15043},[148,149,150,151],[19,15045,77],{"className":15046},[155,236,151],", and ",[19,15049,15051,15070],{"className":15050,"translate":23},[27],[19,15052,15054],{"className":15053},[31],[33,15055,15056],{"xmlns":35},[38,15057,15058,15067],{},[41,15059,15060],{},[65,15061,15062,15065],{},[58,15063,15064],{},"e",[58,15066,77],{},[91,15068,15069],{"encoding":93},"e^n",[19,15071,15073],{"className":15072,"ariaHidden":99},[98],[19,15074,15076,15079],{"className":15075},[103],[19,15077],{"className":15078,"style":15017},[107],[19,15080,15082,15085],{"className":15081},[155],[19,15083,15064],{"className":15084},[155,236],[19,15086,15088],{"className":15087},[122],[19,15089,15091],{"className":15090},[126],[19,15092,15094],{"className":15093},[131],[19,15095,15097],{"className":15096,"style":15017},[135],[19,15098,15099,15102],{"style":12742},[19,15100],{"className":15101,"style":298},[143],[19,15103,15105],{"className":15104},[148,149,150,151],[19,15106,77],{"className":15107},[155,236,151],".",[15110,15111,15112],"blockquote",{},[15,15113,15114,15115,15143,15144,15177,15178,15108],{},"As ",[19,15116,15118,15131],{"className":15117,"translate":23},[27],[19,15119,15121],{"className":15120},[31],[33,15122,15123],{"xmlns":35},[38,15124,15125,15129],{},[41,15126,15127],{},[58,15128,77],{},[91,15130,77],{"encoding":93},[19,15132,15134],{"className":15133,"ariaHidden":99},[98],[19,15135,15137,15140],{"className":15136},[103],[19,15138],{"className":15139,"style":4230},[107],[19,15141,77],{"className":15142},[155,236]," approaches infinity, ",[19,15145,15147,15162],{"className":15146,"translate":23},[27],[19,15148,15150],{"className":15149},[31],[33,15151,15152],{"xmlns":35},[38,15153,15154,15160],{},[41,15155,15156,15158],{},[58,15157,77],{},[47,15159,7425],{"stretchy":411},[91,15161,14970],{"encoding":93},[19,15163,15165],{"className":15164,"ariaHidden":99},[98],[19,15166,15168,15171,15174],{"className":15167},[103],[19,15169],{"className":15170,"style":4286},[107],[19,15172,77],{"className":15173},[155,236],[19,15175,7425],{"className":15176},[255]," is approximately ",[19,15179,15181,15220],{"className":15180,"translate":23},[27],[19,15182,15184],{"className":15183},[31],[33,15185,15186],{"xmlns":35},[38,15187,15188,15217],{},[41,15189,15190,15201],{},[15191,15192,15193],"msqrt",{},[41,15194,15195,15197,15199],{},[51,15196,63],{},[58,15198,60],{},[58,15200,77],{},[65,15202,15203,15215],{},[41,15204,15205,15207,15213],{},[47,15206,412],{"fence":99},[55,15208,15209,15211],{},[58,15210,77],{},[58,15212,15064],{},[47,15214,422],{"fence":99},[58,15216,77],{},[91,15218,15219],{"encoding":93},"\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[19,15221,15223],{"className":15222,"ariaHidden":99},[98],[19,15224,15226,15230,15296,15299],{"className":15225},[103],[19,15227],{"className":15228,"style":15229},[107],"height:1.2572em;vertical-align:-0.35em;",[19,15231,15234],{"className":15232},[155,15233],"sqrt",[19,15235,15237,15287],{"className":15236},[126,127],[19,15238,15240,15284],{"className":15239},[131],[19,15241,15244,15264],{"className":15242,"style":15243},[135],"height:0.9072em;",[19,15245,15248,15251],{"className":15246,"style":10678},[15247],"svg-align",[19,15249],{"className":15250,"style":144},[143],[19,15252,15255,15258,15261],{"className":15253,"style":15254},[155],"padding-left:0.833em;",[19,15256,63],{"className":15257},[155],[19,15259,60],{"className":15260,"style":237},[155,236],[19,15262,77],{"className":15263},[155,236],[19,15265,15267,15270],{"style":15266},"top:-2.8672em;",[19,15268],{"className":15269,"style":144},[143],[19,15271,15275],{"className":15272,"style":15274},[15273],"hide-tail","min-width:0.853em;height:1.08em;",[1459,15276,15281],{"xmlns":1461,"width":15277,"height":15278,"viewBox":15279,"preserveAspectRatio":15280},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[1466,15282],{"d":15283},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[19,15285,242],{"className":15286},[241],[19,15288,15290],{"className":15289},[131],[19,15291,15294],{"className":15292,"style":15293},[135],"height:0.1328em;",[19,15295],{},[19,15297],{"className":15298,"style":272},[271],[19,15300,15302,15385],{"className":15301},[1353],[19,15303,15305,15311,15379],{"className":15304},[1353],[19,15306,15308],{"className":15307,"style":1358},[177,1357],[19,15309,412],{"className":15310},[1362,203],[19,15312,15314,15317,15376],{"className":15313},[155],[19,15315],{"className":15316},[177,178],[19,15318,15320],{"className":15319},[55],[19,15321,15323,15368],{"className":15322},[126,127],[19,15324,15326,15365],{"className":15325},[131],[19,15327,15329,15343,15351],{"className":15328,"style":8761},[135],[19,15330,15331,15334],{"style":8764},[19,15332],{"className":15333,"style":144},[143],[19,15335,15337],{"className":15336},[148,149,150,151],[19,15338,15340],{"className":15339},[155,151],[19,15341,15064],{"className":15342},[155,236,151],[19,15344,15345,15348],{"style":3968},[19,15346],{"className":15347,"style":144},[143],[19,15349],{"className":15350,"style":3975},[219],[19,15352,15353,15356],{"style":8787},[19,15354],{"className":15355,"style":144},[143],[19,15357,15359],{"className":15358},[148,149,150,151],[19,15360,15362],{"className":15361},[155,151],[19,15363,77],{"className":15364},[155,236,151],[19,15366,242],{"className":15367},[241],[19,15369,15371],{"className":15370},[131],[19,15372,15374],{"className":15373,"style":8809},[135],[19,15375],{},[19,15377],{"className":15378},[255,178],[19,15380,15382],{"className":15381,"style":1358},[255,1357],[19,15383,422],{"className":15384},[1362,203],[19,15386,15388],{"className":15387},[122],[19,15389,15391],{"className":15390},[126],[19,15392,15394],{"className":15393},[131],[19,15395,15398],{"className":15396,"style":15397},[135],"height:0.9043em;",[19,15399,15401,15404],{"style":15400},"top:-3.3029em;margin-right:0.05em;",[19,15402],{"className":15403,"style":298},[143],[19,15405,15407],{"className":15406},[148,149,150,151],[19,15408,77],{"className":15409},[155,236,151],[15,15411,15412],{},"First, introduce a limit:",[19,15414,15416],{"className":15415,"translate":23},[22],[19,15417,15419,15483],{"className":15418,"translate":23},[27],[19,15420,15422],{"className":15421},[31],[33,15423,15424],{"xmlns":35,"display":36},[38,15425,15426,15480],{},[41,15427,15428,15444,15472,15474],{},[10327,15429,15430,15436],{},[41,15431,15432,15434],{},[58,15433,10333],{},[47,15435,74],{},[41,15437,15438,15440,15442],{},[58,15439,77],{},[47,15441,10342],{},[58,15443,10346],{"mathvariant":10345},[65,15445,15446,15466],{},[41,15447,15448,15450,15464],{},[47,15449,412],{"fence":99},[55,15451,15452,15458],{},[41,15453,15454,15456],{},[58,15455,77],{},[47,15457,7425],{"stretchy":411},[65,15459,15460,15462],{},[58,15461,77],{},[58,15463,77],{},[47,15465,422],{"fence":99},[55,15467,15468,15470],{},[51,15469,402],{},[58,15471,77],{},[47,15473,370],{},[55,15475,15476,15478],{},[51,15477,402],{},[58,15479,15064],{},[91,15481,15482],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\frac{1}{e}",[19,15484,15486,15764],{"className":15485,"ariaHidden":99},[98],[19,15487,15489,15493,15548,15551,15755,15758,15761],{"className":15488},[103],[19,15490],{"className":15491,"style":15492},[107],"height:2.744em;vertical-align:-0.95em;",[19,15494,15496],{"className":15495},[112,10645],[19,15497,15499,15539],{"className":15498},[126,127],[19,15500,15502,15536],{"className":15501},[131],[19,15503,15505,15526],{"className":15504,"style":4286},[135],[19,15506,15508,15511],{"style":15507},"top:-2.4em;margin-left:0em;",[19,15509],{"className":15510,"style":144},[143],[19,15512,15514],{"className":15513},[148,149,150,151],[19,15515,15517,15520,15523],{"className":15516},[155,151],[19,15518,77],{"className":15519},[155,236,151],[19,15521,10342],{"className":15522},[611,151],[19,15524,10346],{"className":15525},[155,151],[19,15527,15528,15531],{"style":10678},[19,15529],{"className":15530,"style":144},[143],[19,15532,15533],{},[19,15534,10333],{"className":15535},[112],[19,15537,242],{"className":15538},[241],[19,15540,15542],{"className":15541},[131],[19,15543,15546],{"className":15544,"style":15545},[135],"height:0.7em;",[19,15547],{},[19,15549],{"className":15550,"style":272},[271],[19,15552,15554,15661],{"className":15553},[1353],[19,15555,15557,15563,15655],{"className":15556},[1353],[19,15558,15560],{"className":15559,"style":1358},[177,1357],[19,15561,412],{"className":15562},[1362,150],[19,15564,15566,15569,15652],{"className":15565},[155],[19,15567],{"className":15568},[177,178],[19,15570,15572],{"className":15571},[55],[19,15573,15575,15644],{"className":15574},[126,127],[19,15576,15578,15641],{"className":15577},[131],[19,15579,15581,15619,15627],{"className":15580,"style":4771},[135],[19,15582,15583,15586],{"style":3956},[19,15584],{"className":15585,"style":144},[143],[19,15587,15589],{"className":15588},[155],[19,15590,15592,15595],{"className":15591},[155],[19,15593,77],{"className":15594},[155,236],[19,15596,15598],{"className":15597},[122],[19,15599,15601],{"className":15600},[126],[19,15602,15604],{"className":15603},[131],[19,15605,15608],{"className":15606,"style":15607},[135],"height:0.5904em;",[19,15609,15610,15613],{"style":12929},[19,15611],{"className":15612,"style":298},[143],[19,15614,15616],{"className":15615},[148,149,150,151],[19,15617,77],{"className":15618},[155,236,151],[19,15620,15621,15624],{"style":3968},[19,15622],{"className":15623,"style":144},[143],[19,15625],{"className":15626,"style":3975},[219],[19,15628,15629,15632],{"style":3978},[19,15630],{"className":15631,"style":144},[143],[19,15633,15635,15638],{"className":15634},[155],[19,15636,77],{"className":15637},[155,236],[19,15639,7425],{"className":15640},[255],[19,15642,242],{"className":15643},[241],[19,15645,15647],{"className":15646},[131],[19,15648,15650],{"className":15649,"style":4009},[135],[19,15651],{},[19,15653],{"className":15654},[255,178],[19,15656,15658],{"className":15657,"style":1358},[255,1357],[19,15659,422],{"className":15660},[1362,150],[19,15662,15664],{"className":15663},[122],[19,15665,15667],{"className":15666},[126],[19,15668,15670],{"className":15669},[131],[19,15671,15674],{"className":15672,"style":15673},[135],"height:1.7939em;",[19,15675,15677,15680],{"style":15676},"top:-4.2029em;margin-right:0.05em;",[19,15678],{"className":15679,"style":144},[143],[19,15681,15683],{"className":15682},[148,149,150,151],[19,15684,15686],{"className":15685},[155,151],[19,15687,15689,15692,15752],{"className":15688},[155,151],[19,15690],{"className":15691},[177,178,148,179,180],[19,15693,15695],{"className":15694},[55],[19,15696,15698,15744],{"className":15697},[126,127],[19,15699,15701,15741],{"className":15700},[131],[19,15702,15705,15719,15727],{"className":15703,"style":15704},[135],"height:0.8443em;",[19,15706,15707,15710],{"style":196},[19,15708],{"className":15709,"style":144},[143],[19,15711,15713],{"className":15712},[148,179,203,151],[19,15714,15716],{"className":15715},[155,151],[19,15717,77],{"className":15718},[155,236,151],[19,15720,15721,15724],{"style":212},[19,15722],{"className":15723,"style":144},[143],[19,15725],{"className":15726,"style":220},[219,151],[19,15728,15729,15732],{"style":223},[19,15730],{"className":15731,"style":144},[143],[19,15733,15735],{"className":15734},[148,179,203,151],[19,15736,15738],{"className":15737},[155,151],[19,15739,402],{"className":15740},[155,151],[19,15742,242],{"className":15743},[241],[19,15745,15747],{"className":15746},[131],[19,15748,15750],{"className":15749,"style":249},[135],[19,15751],{},[19,15753],{"className":15754},[255,178,148,179,180],[19,15756],{"className":15757,"style":607},[271],[19,15759,370],{"className":15760},[611],[19,15762],{"className":15763,"style":607},[271],[19,15765,15767,15771],{"className":15766},[103],[19,15768],{"className":15769,"style":15770},[107],"height:2.0074em;vertical-align:-0.686em;",[19,15772,15774,15777,15830],{"className":15773},[155],[19,15775],{"className":15776},[177,178],[19,15778,15780],{"className":15779},[55],[19,15781,15783,15822],{"className":15782},[126,127],[19,15784,15786,15819],{"className":15785},[131],[19,15787,15789,15800,15808],{"className":15788,"style":3953},[135],[19,15790,15791,15794],{"style":3956},[19,15792],{"className":15793,"style":144},[143],[19,15795,15797],{"className":15796},[155],[19,15798,15064],{"className":15799},[155,236],[19,15801,15802,15805],{"style":3968},[19,15803],{"className":15804,"style":144},[143],[19,15806],{"className":15807,"style":3975},[219],[19,15809,15810,15813],{"style":3978},[19,15811],{"className":15812,"style":144},[143],[19,15814,15816],{"className":15815},[155],[19,15817,402],{"className":15818},[155],[19,15820,242],{"className":15821},[241],[19,15823,15825],{"className":15824},[131],[19,15826,15828],{"className":15827,"style":4009},[135],[19,15829],{},[19,15831],{"className":15832},[255,178],[15,15834,15835],{},"Proof:",[19,15837,15839],{"className":15838,"translate":23},[22],[19,15840,15842,15953],{"className":15841,"translate":23},[27],[19,15843,15845],{"className":15844},[31],[33,15846,15847],{"xmlns":35,"display":36},[38,15848,15849,15950],{},[41,15850,15851,15867,15895,15897,15913],{},[10327,15852,15853,15859],{},[41,15854,15855,15857],{},[58,15856,10333],{},[47,15858,74],{},[41,15860,15861,15863,15865],{},[58,15862,77],{},[47,15864,10342],{},[58,15866,10346],{"mathvariant":10345},[65,15868,15869,15889],{},[41,15870,15871,15873,15887],{},[47,15872,412],{"fence":99},[55,15874,15875,15881],{},[41,15876,15877,15879],{},[58,15878,77],{},[47,15880,7425],{"stretchy":411},[65,15882,15883,15885],{},[58,15884,77],{},[58,15886,77],{},[47,15888,422],{"fence":99},[55,15890,15891,15893],{},[51,15892,402],{},[58,15894,77],{},[47,15896,370],{},[10327,15898,15899,15905],{},[41,15900,15901,15903],{},[58,15902,10333],{},[47,15904,74],{},[41,15906,15907,15909,15911],{},[58,15908,77],{},[47,15910,10342],{},[58,15912,10346],{"mathvariant":10345},[65,15914,15915,15917],{},[58,15916,15064],{},[41,15918,15919,15925,15928,15930],{},[55,15920,15921,15923],{},[51,15922,402],{},[58,15924,77],{},[58,15926,15927],{},"ln",[47,15929,74],{},[41,15931,15932,15934,15948],{},[47,15933,412],{"fence":99},[55,15935,15936,15942],{},[41,15937,15938,15940],{},[58,15939,77],{},[47,15941,7425],{"stretchy":411},[65,15943,15944,15946],{},[58,15945,77],{},[58,15947,77],{},[47,15949,422],{"fence":99},[91,15951,15952],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\lim_{n \\rightarrow \\infty} e^{\\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right)}",[19,15954,15956,16227],{"className":15955,"ariaHidden":99},[98],[19,15957,15959,15962,16015,16018,16218,16221,16224],{"className":15958},[103],[19,15960],{"className":15961,"style":15492},[107],[19,15963,15965],{"className":15964},[112,10645],[19,15966,15968,16007],{"className":15967},[126,127],[19,15969,15971,16004],{"className":15970},[131],[19,15972,15974,15994],{"className":15973,"style":4286},[135],[19,15975,15976,15979],{"style":15507},[19,15977],{"className":15978,"style":144},[143],[19,15980,15982],{"className":15981},[148,149,150,151],[19,15983,15985,15988,15991],{"className":15984},[155,151],[19,15986,77],{"className":15987},[155,236,151],[19,15989,10342],{"className":15990},[611,151],[19,15992,10346],{"className":15993},[155,151],[19,15995,15996,15999],{"style":10678},[19,15997],{"className":15998,"style":144},[143],[19,16000,16001],{},[19,16002,10333],{"className":16003},[112],[19,16005,242],{"className":16006},[241],[19,16008,16010],{"className":16009},[131],[19,16011,16013],{"className":16012,"style":15545},[135],[19,16014],{},[19,16016],{"className":16017,"style":272},[271],[19,16019,16021,16127],{"className":16020},[1353],[19,16022,16024,16030,16121],{"className":16023},[1353],[19,16025,16027],{"className":16026,"style":1358},[177,1357],[19,16028,412],{"className":16029},[1362,150],[19,16031,16033,16036,16118],{"className":16032},[155],[19,16034],{"className":16035},[177,178],[19,16037,16039],{"className":16038},[55],[19,16040,16042,16110],{"className":16041},[126,127],[19,16043,16045,16107],{"className":16044},[131],[19,16046,16048,16085,16093],{"className":16047,"style":4771},[135],[19,16049,16050,16053],{"style":3956},[19,16051],{"className":16052,"style":144},[143],[19,16054,16056],{"className":16055},[155],[19,16057,16059,16062],{"className":16058},[155],[19,16060,77],{"className":16061},[155,236],[19,16063,16065],{"className":16064},[122],[19,16066,16068],{"className":16067},[126],[19,16069,16071],{"className":16070},[131],[19,16072,16074],{"className":16073,"style":15607},[135],[19,16075,16076,16079],{"style":12929},[19,16077],{"className":16078,"style":298},[143],[19,16080,16082],{"className":16081},[148,149,150,151],[19,16083,77],{"className":16084},[155,236,151],[19,16086,16087,16090],{"style":3968},[19,16088],{"className":16089,"style":144},[143],[19,16091],{"className":16092,"style":3975},[219],[19,16094,16095,16098],{"style":3978},[19,16096],{"className":16097,"style":144},[143],[19,16099,16101,16104],{"className":16100},[155],[19,16102,77],{"className":16103},[155,236],[19,16105,7425],{"className":16106},[255],[19,16108,242],{"className":16109},[241],[19,16111,16113],{"className":16112},[131],[19,16114,16116],{"className":16115,"style":4009},[135],[19,16117],{},[19,16119],{"className":16120},[255,178],[19,16122,16124],{"className":16123,"style":1358},[255,1357],[19,16125,422],{"className":16126},[1362,150],[19,16128,16130],{"className":16129},[122],[19,16131,16133],{"className":16132},[126],[19,16134,16136],{"className":16135},[131],[19,16137,16139],{"className":16138,"style":15673},[135],[19,16140,16141,16144],{"style":15676},[19,16142],{"className":16143,"style":144},[143],[19,16145,16147],{"className":16146},[148,149,150,151],[19,16148,16150],{"className":16149},[155,151],[19,16151,16153,16156,16215],{"className":16152},[155,151],[19,16154],{"className":16155},[177,178,148,179,180],[19,16157,16159],{"className":16158},[55],[19,16160,16162,16207],{"className":16161},[126,127],[19,16163,16165,16204],{"className":16164},[131],[19,16166,16168,16182,16190],{"className":16167,"style":15704},[135],[19,16169,16170,16173],{"style":196},[19,16171],{"className":16172,"style":144},[143],[19,16174,16176],{"className":16175},[148,179,203,151],[19,16177,16179],{"className":16178},[155,151],[19,16180,77],{"className":16181},[155,236,151],[19,16183,16184,16187],{"style":212},[19,16185],{"className":16186,"style":144},[143],[19,16188],{"className":16189,"style":220},[219,151],[19,16191,16192,16195],{"style":223},[19,16193],{"className":16194,"style":144},[143],[19,16196,16198],{"className":16197},[148,179,203,151],[19,16199,16201],{"className":16200},[155,151],[19,16202,402],{"className":16203},[155,151],[19,16205,242],{"className":16206},[241],[19,16208,16210],{"className":16209},[131],[19,16211,16213],{"className":16212,"style":249},[135],[19,16214],{},[19,16216],{"className":16217},[255,178,148,179,180],[19,16219],{"className":16220,"style":607},[271],[19,16222,370],{"className":16223},[611],[19,16225],{"className":16226,"style":607},[271],[19,16228,16230,16234,16287,16290],{"className":16229},[103],[19,16231],{"className":16232,"style":16233},[107],"height:1.8105em;vertical-align:-0.7em;",[19,16235,16237],{"className":16236},[112,10645],[19,16238,16240,16279],{"className":16239},[126,127],[19,16241,16243,16276],{"className":16242},[131],[19,16244,16246,16266],{"className":16245,"style":4286},[135],[19,16247,16248,16251],{"style":15507},[19,16249],{"className":16250,"style":144},[143],[19,16252,16254],{"className":16253},[148,149,150,151],[19,16255,16257,16260,16263],{"className":16256},[155,151],[19,16258,77],{"className":16259},[155,236,151],[19,16261,10342],{"className":16262},[611,151],[19,16264,10346],{"className":16265},[155,151],[19,16267,16268,16271],{"style":10678},[19,16269],{"className":16270,"style":144},[143],[19,16272,16273],{},[19,16274,10333],{"className":16275},[112],[19,16277,242],{"className":16278},[241],[19,16280,16282],{"className":16281},[131],[19,16283,16285],{"className":16284,"style":15545},[135],[19,16286],{},[19,16288],{"className":16289,"style":272},[271],[19,16291,16293,16296],{"className":16292},[155],[19,16294,15064],{"className":16295},[155,236],[19,16297,16299],{"className":16298},[122],[19,16300,16302],{"className":16301},[126],[19,16303,16305],{"className":16304},[131],[19,16306,16309],{"className":16307,"style":16308},[135],"height:1.1105em;",[19,16310,16312,16315],{"style":16311},"top:-3.413em;margin-right:0.05em;",[19,16313],{"className":16314,"style":144},[143],[19,16316,16318],{"className":16317},[148,149,150,151],[19,16319,16321,16389,16393,16403],{"className":16320},[155,151],[19,16322,16324,16327,16386],{"className":16323},[155,151],[19,16325],{"className":16326},[177,178,148,179,180],[19,16328,16330],{"className":16329},[55],[19,16331,16333,16378],{"className":16332},[126,127],[19,16334,16336,16375],{"className":16335},[131],[19,16337,16339,16353,16361],{"className":16338,"style":15704},[135],[19,16340,16341,16344],{"style":196},[19,16342],{"className":16343,"style":144},[143],[19,16345,16347],{"className":16346},[148,179,203,151],[19,16348,16350],{"className":16349},[155,151],[19,16351,77],{"className":16352},[155,236,151],[19,16354,16355,16358],{"style":212},[19,16356],{"className":16357,"style":144},[143],[19,16359],{"className":16360,"style":220},[219,151],[19,16362,16363,16366],{"style":223},[19,16364],{"className":16365,"style":144},[143],[19,16367,16369],{"className":16368},[148,179,203,151],[19,16370,16372],{"className":16371},[155,151],[19,16373,402],{"className":16374},[155,151],[19,16376,242],{"className":16377},[241],[19,16379,16381],{"className":16380},[131],[19,16382,16384],{"className":16383,"style":249},[135],[19,16385],{},[19,16387],{"className":16388},[255,178,148,179,180],[19,16390],{"className":16391,"style":16392},[271,151],"margin-right:0.1952em;",[19,16394,16396,16400],{"className":16395},[112,151],[19,16397,16399],{"className":16398},[151],"l",[19,16401,77],{"className":16402},[151],[19,16404,16406,16413,16511],{"className":16405},[1353,151],[19,16407,16410],{"className":16408,"style":16409},[177,148,179,180,151,1357],"top:0.075em;",[19,16411,412],{"className":16412},[151],[19,16414,16416,16419,16508],{"className":16415},[155,151],[19,16417],{"className":16418},[177,178,148,179,180],[19,16420,16422],{"className":16421},[55],[19,16423,16425,16500],{"className":16424},[126,127],[19,16426,16428,16497],{"className":16427},[131],[19,16429,16432,16472,16480],{"className":16430,"style":16431},[135],"height:0.88em;",[19,16433,16434,16437],{"style":196},[19,16435],{"className":16436,"style":144},[143],[19,16438,16440],{"className":16439},[148,179,203,151],[19,16441,16443],{"className":16442},[155,151],[19,16444,16446,16449],{"className":16445},[155,151],[19,16447,77],{"className":16448},[155,236,151],[19,16450,16452],{"className":16451},[122],[19,16453,16455],{"className":16454},[126],[19,16456,16458],{"className":16457},[131],[19,16459,16462],{"className":16460,"style":16461},[135],"height:0.7246em;",[19,16463,16465,16469],{"style":16464},"top:-2.794em;margin-right:0.1em;",[19,16466],{"className":16467,"style":16468},[143],"height:2.5em;",[19,16470,77],{"className":16471},[155,236,151],[19,16473,16474,16477],{"style":212},[19,16475],{"className":16476,"style":144},[143],[19,16478],{"className":16479,"style":220},[219,151],[19,16481,16482,16485],{"style":223},[19,16483],{"className":16484,"style":144},[143],[19,16486,16488],{"className":16487},[148,179,203,151],[19,16489,16491,16494],{"className":16490},[155,151],[19,16492,77],{"className":16493},[155,236,151],[19,16495,7425],{"className":16496},[255,151],[19,16498,242],{"className":16499},[241],[19,16501,16503],{"className":16502},[131],[19,16504,16506],{"className":16505,"style":249},[135],[19,16507],{},[19,16509],{"className":16510},[255,178,148,179,180],[19,16512,16514],{"className":16513,"style":16409},[255,148,179,180,151,1357],[19,16515,422],{"className":16516},[151],[15,16518,16519],{},"which is equivalent to proving:",[19,16521,16523],{"className":16522,"translate":23},[22],[19,16524,16526,16590],{"className":16525,"translate":23},[27],[19,16527,16529],{"className":16528},[31],[33,16530,16531],{"xmlns":35,"display":36},[38,16532,16533,16587],{},[41,16534,16535,16551,16557,16559,16561,16581,16583,16585],{},[10327,16536,16537,16543],{},[41,16538,16539,16541],{},[58,16540,10333],{},[47,16542,74],{},[41,16544,16545,16547,16549],{},[58,16546,77],{},[47,16548,10342],{},[58,16550,10346],{"mathvariant":10345},[55,16552,16553,16555],{},[51,16554,402],{},[58,16556,77],{},[58,16558,15927],{},[47,16560,74],{},[41,16562,16563,16565,16579],{},[47,16564,412],{"fence":99},[55,16566,16567,16573],{},[41,16568,16569,16571],{},[58,16570,77],{},[47,16572,7425],{"stretchy":411},[65,16574,16575,16577],{},[58,16576,77],{},[58,16578,77],{},[47,16580,422],{"fence":99},[47,16582,370],{},[47,16584,373],{},[51,16586,402],{},[91,16588,16589],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) = 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have:",[19,16858,16860],{"className":16859,"translate":23},[22],[19,16861,16863,17047],{"className":16862,"translate":23},[27],[19,16864,16866],{"className":16865},[31],[33,16867,16868],{"xmlns":35,"display":36},[38,16869,16870,17044],{},[41,16871,16872,16888,16894,16896,16898,16918,16920,16922,16938,16944,16988,16990,16992,17000,17002,17004,17006,17008,17010,17012,17014,17016,17018,17020,17022,17024,17026,17028,17030,17038,17040,17042],{},[10327,16873,16874,16880],{},[41,16875,16876,16878],{},[58,16877,10333],{},[47,16879,74],{},[41,16881,16882,16884,16886],{},[58,16883,77],{},[47,16885,10342],{},[58,16887,10346],{"mathvariant":10345},[55,16889,16890,16892],{},[51,16891,402],{},[58,16893,77],{},[58,16895,15927],{},[47,16897,74],{},[41,16899,16900,16902,16916],{},[47,16901,412],{"fence":99},[55,16903,16904,16910],{},[41,16905,16906,16908],{},[58,16907,77],{},[47,16909,7425],{"stretchy":411},[65,16911,16912,16914],{},[58,16913,77],{},[58,16915,77],{},[47,16917,422],{"fence":99},[271,16919],{"linebreak":1015},[47,16921,370],{},[10327,16923,16924,16930],{},[41,16925,16926,16928],{},[58,16927,10333],{},[47,16929,74],{},[41,16931,16932,16934,16936],{},[58,16933,77],{},[47,16935,10342],{},[58,16937,10346],{"mathvariant":10345},[55,16939,16940,16942],{},[51,16941,402],{},[58,16943,77],{},[41,16945,16946,16948,16950,16952,16958,16960,16962,16964,16970,16972,16974,16976,16978,16980,16986],{},[47,16947,412],{"fence":99},[58,16949,15927],{},[47,16951,74],{},[55,16953,16954,16956],{},[51,16955,402],{},[58,16957,77],{},[47,16959,1066],{},[58,16961,15927],{},[47,16963,74],{},[55,16965,16966,16968],{},[51,16967,63],{},[58,16969,77],{},[47,16971,1066],{},[47,16973,4451],{},[47,16975,1066],{},[58,16977,15927],{},[47,16979,74],{},[55,16981,16982,16984],{},[58,16983,77],{},[58,16985,77],{},[47,16987,422],{"fence":99},[271,16989],{"linebreak":1015},[47,16991,370],{},[44,16993,16994,16996,16998],{},[47,16995,49],{},[51,16997,53],{},[51,16999,402],{},[58,17001,15927],{},[47,17003,74],{},[58,17005,80],{},[82,17007,84],{},[58,17009,87],{},[58,17011,80],{},[47,17013,370],{},[47,17015,412],{"stretchy":411},[58,17017,80],{},[58,17019,15927],{},[47,17021,74],{},[58,17023,80],{},[47,17025,373],{},[58,17027,80],{},[47,17029,422],{"stretchy":411},[44,17031,17032,17034,17036],{},[47,17033,1055],{"fence":411,"stretchy":99,"minsize":1054,"maxsize":1054},[51,17035,53],{},[51,17037,402],{},[47,17039,370],{},[47,17041,373],{},[51,17043,402],{},[91,17045,17046],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) \\\\ = \\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\left( \\ln \\frac{1}{n} + \\ln \\frac{2}{n} + \\cdots + \\ln \\frac{n}{n} \\right) \\\\ = \\int_{0}^{1} \\ln x \\, dx = (x \\ln x - x) \\Big|_{0}^{1} = 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would like to verify:",[19,18211,18213],{"className":18212,"translate":23},[22],[19,18214,18216,18268],{"className":18215,"translate":23},[27],[19,18217,18219],{"className":18218},[31],[33,18220,18221],{"xmlns":35,"display":36},[38,18222,18223,18265],{},[41,18224,18225,18241,18255,18261,18263],{},[10327,18226,18227,18233],{},[41,18228,18229,18231],{},[58,18230,10333],{},[47,18232,74],{},[41,18234,18235,18237,18239],{},[58,18236,77],{},[47,18238,10342],{},[58,18240,10346],{"mathvariant":10345},[55,18242,18243,18249],{},[41,18244,18245,18247],{},[58,18246,77],{},[47,18248,7425],{"stretchy":411},[65,18250,18251,18253],{},[58,18252,77],{},[58,18254,77],{},[65,18256,18257,18259],{},[58,18258,15064],{},[58,18260,77],{},[47,18262,370],{},[51,18264,402],{},[91,18266,18267],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^n} e^n = 1",[19,18269,18271,18464],{"className":18270,"ariaHidden":99},[98],[19,18272,18274,18278,18331,18334,18425,18455,18458,18461],{"className":18273},[103],[19,18275],{"className":18276,"style":18277},[107],"height:2.0714em;vertical-align:-0.7em;",[19,18279,18281],{"className":18280},[112,10645],[19,18282,18284,18323],{"className":18283},[126,127],[19,18285,18287,18320],{"className":18286},[131],[19,18288,18290,18310],{"className":18289,"style":4286},[135],[19,18291,18292,18295],{"style":15507},[19,18293],{"className":18294,"style":144},[143],[19,18296,18298],{"className":18297},[148,149,150,151],[19,18299,18301,18304,18307],{"className":18300},[155,151],[19,18302,77],{"className":18303},[155,236,151],[19,18305,10342],{"className":18306},[611,151],[19,18308,10346],{"className":18309},[155,151],[19,18311,18312,18315],{"style":10678},[19,18313],{"className":18314,"style":144},[143],[19,18316,18317],{},[19,18318,10333],{"className":18319},[112],[19,18321,242],{"className":18322},[241],[19,18324,18326],{"className":18325},[131],[19,18327,18329],{"className":18328,"style":15545},[135],[19,18330],{},[19,18332],{"className":18333,"style":272},[271],[19,18335,18337,18340,18422],{"className":18336},[155],[19,18338],{"className":18339},[177,178],[19,18341,18343],{"className":18342},[55],[19,18344,18346,18414],{"className":18345},[126,127],[19,18347,18349,18411],{"className":18348},[131],[19,18350,18352,18389,18397],{"className":18351,"style":4771},[135],[19,18353,18354,18357],{"style":3956},[19,18355],{"className":18356,"style":144},[143],[19,18358,18360],{"className":18359},[155],[19,18361,18363,18366],{"className":18362},[155],[19,18364,77],{"className":18365},[155,236],[19,18367,18369],{"className":18368},[122],[19,18370,18372],{"className":18371},[126],[19,18373,18375],{"className":18374},[131],[19,18376,18378],{"className":18377,"style":15607},[135],[19,18379,18380,18383],{"style":12929},[19,18381],{"className":18382,"style":298},[143],[19,18384,18386],{"className":18385},[148,149,150,151],[19,18387,77],{"className":18388},[155,236,151],[19,18390,18391,18394],{"style":3968},[19,18392],{"className":18393,"style":144},[143],[19,18395],{"className":18396,"style":3975},[219],[19,18398,18399,18402],{"style":3978},[19,18400],{"className":18401,"style":144},[143],[19,18403,18405,18408],{"className":18404},[155],[19,18406,77],{"className":18407},[155,236],[19,18409,7425],{"className":18410},[255],[19,18412,242],{"className":18413},[241],[19,18415,18417],{"className":18416},[131],[19,18418,18420],{"className":18419,"style":4009},[135],[19,18421],{},[19,18423],{"className":18424},[255,178],[19,18426,18428,18431],{"className":18427},[155],[19,18429,15064],{"className":18430},[155,236],[19,18432,18434],{"className":18433},[122],[19,18435,18437],{"className":18436},[126],[19,18438,18440],{"className":18439},[131],[19,18441,18444],{"className":18442,"style":18443},[135],"height:0.7144em;",[19,18445,18446,18449],{"style":2136},[19,18447],{"className":18448,"style":298},[143],[19,18450,18452],{"className":18451},[148,149,150,151],[19,18453,77],{"className":18454},[155,236,151],[19,18456],{"className":18457,"style":607},[271],[19,18459,370],{"className":18460},[611],[19,18462],{"className":18463,"style":607},[271],[19,18465,18467,18470],{"className":18466},[103],[19,18468],{"className":18469,"style":5939},[107],[19,18471,402],{"className":18472},[155],[15,18474,18475],{},"Let’s test this.",[15,18477,18478],{},[4202,18479,18480],{},"Using the ratio test:",[19,18482,18484],{"className":18483,"translate":23},[22],[19,18485,18487,18598],{"className":18486,"translate":23},[27],[19,18488,18490],{"className":18489},[31],[33,18491,18492],{"xmlns":35,"display":36},[38,18493,18494,18595],{},[41,18495,18496,18504,18506,18520,18526,18528,18531,18543,18545,18583],{},[18497,18498,18499,18502],"msub",{},[58,18500,18501],{},"a",[58,18503,77],{},[47,18505,370],{},[55,18507,18508,18514],{},[41,18509,18510,18512],{},[58,18511,77],{},[47,18513,7425],{"stretchy":411},[65,18515,18516,18518],{},[58,18517,77],{},[58,18519,77],{},[65,18521,18522,18524],{},[58,18523,15064],{},[58,18525,77],{},[47,18527,8689],{"separator":99},[271,18529],{"width":18530},"1em",[18497,18532,18533,18535],{},[58,18534,18501],{},[41,18536,18537,18539,18541],{},[58,18538,77],{},[47,18540,1066],{},[51,18542,402],{},[47,18544,370],{},[55,18546,18547,18561],{},[41,18548,18549,18551,18553,18555,18557,18559],{},[47,18550,412],{"stretchy":411},[58,18552,77],{},[47,18554,1066],{},[51,18556,402],{},[47,18558,422],{"stretchy":411},[47,18560,7425],{"stretchy":411},[41,18562,18563,18565,18567,18569,18571],{},[47,18564,412],{"stretchy":411},[58,18566,77],{},[47,18568,1066],{},[51,18570,402],{},[65,18572,18573,18575],{},[47,18574,422],{"stretchy":411},[41,18576,18577,18579,18581],{},[58,18578,77],{},[47,18580,1066],{},[51,18582,402],{},[65,18584,18585,18587],{},[58,18586,15064],{},[41,18588,18589,18591,18593],{},[58,18590,77],{},[47,18592,1066],{},[51,18594,402],{},[91,18596,18597],{"encoding":93},"a_n = \\frac{n!}{n^n} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n+1}} e^{n+1}",[19,18599,18601,18660,18856],{"className":18600,"ariaHidden":99},[98],[19,18602,18604,18608,18651,18654,18657],{"className":18603},[103],[19,18605],{"className":18606,"style":18607},[107],"height:0.5806em;vertical-align:-0.15em;",[19,18609,18611,18614],{"className":18610},[155],[19,18612,18501],{"className":18613},[155,236],[19,18615,18617],{"className":18616},[122],[19,18618,18620,18642],{"className":18619},[126,127],[19,18621,18623,18639],{"className":18622},[131],[19,18624,18627],{"className":18625,"style":18626},[135],"height:0.1514em;",[19,18628,18630,18633],{"style":18629},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[19,18631],{"className":18632,"style":298},[143],[19,18634,18636],{"className":18635},[148,149,150,151],[19,18637,77],{"className":18638},[155,236,151],[19,18640,242],{"className":18641},[241],[19,18643,18645],{"className":18644},[131],[19,18646,18649],{"className":18647,"style":18648},[135],"height:0.15em;",[19,18650],{},[19,18652],{"className":18653,"style":607},[271],[19,18655,370],{"className":18656},[611],[19,18658],{"className":18659,"style":607},[271],[19,18661,18663,18666,18757,18786,18789,18793,18796,18847,18850,18853],{"className":18662},[103],[19,18664],{"className":18665,"style":5039},[107],[19,18667,18669,18672,18754],{"className":18668},[155],[19,18670],{"className":18671},[177,178],[19,18673,18675],{"className":18674},[55],[19,18676,18678,18746],{"className":18677},[126,127],[19,18679,18681,18743],{"className":18680},[131],[19,18682,18684,18721,18729],{"className":18683,"style":4771},[135],[19,18685,18686,18689],{"style":3956},[19,18687],{"className":18688,"style":144},[143],[19,18690,18692],{"className":18691},[155],[19,18693,18695,18698],{"className":18694},[155],[19,18696,77],{"className":18697},[155,236],[19,18699,18701],{"className":18700},[122],[19,18702,18704],{"className":18703},[126],[19,18705,18707],{"className":18706},[131],[19,18708,18710],{"className":18709,"style":15607},[135],[19,18711,18712,18715],{"style":12929},[19,18713],{"className":18714,"style":298},[143],[19,18716,18718],{"className":18717},[148,149,150,151],[19,18719,77],{"className":18720},[155,236,151],[19,18722,18723,18726],{"style":3968},[19,18724],{"className":18725,"style":144},[143],[19,18727],{"className":18728,"style":3975},[219],[19,18730,18731,18734],{"style":3978},[19,18732],{"className":18733,"style":144},[143],[19,18735,18737,18740],{"className":18736},[155],[19,18738,77],{"className":18739},[155,236],[19,18741,7425],{"className":18742},[255],[19,18744,242],{"className":18745},[241],[19,18747,18749],{"className":18748},[131],[19,18750,18752],{"className":18751,"style":4009},[135],[19,18753],{},[19,18755],{"className":18756},[255,178],[19,18758,18760,18763],{"className":18759},[155],[19,18761,15064],{"className":18762},[155,236],[19,18764,18766],{"className":18765},[122],[19,18767,18769],{"className":18768},[126],[19,18770,18772],{"className":18771},[131],[19,18773,18775],{"className":18774,"style":18443},[135],[19,18776,18777,18780],{"style":2136},[19,18778],{"className":18779,"style":298},[143],[19,18781,18783],{"className":18782},[148,149,150,151],[19,18784,77],{"className":18785},[155,236,151],[19,18787,8689],{"className":18788},[8739],[19,18790],{"className":18791,"style":18792},[271],"margin-right:1em;",[19,18794],{"className":18795,"style":272},[271],[19,18797,18799,18802],{"className":18798},[155],[19,18800,18501],{"className":18801},[155,236],[19,18803,18805],{"className":18804},[122],[19,18806,18808,18838],{"className":18807},[126,127],[19,18809,18811,18835],{"className":18810},[131],[19,18812,18815],{"className":18813,"style":18814},[135],"height:0.3011em;",[19,18816,18817,18820],{"style":18629},[19,18818],{"className":18819,"style":298},[143],[19,18821,18823],{"className":18822},[148,149,150,151],[19,18824,18826,18829,18832],{"className":18825},[155,151],[19,18827,77],{"className":18828},[155,236,151],[19,18830,1066],{"className":18831},[788,151],[19,18833,402],{"className":18834},[155,151],[19,18836,242],{"className":18837},[241],[19,18839,18841],{"className":18840},[131],[19,18842,18845],{"className":18843,"style":18844},[135],"height:0.2083em;",[19,18846],{},[19,18848],{"className":18849,"style":607},[271],[19,18851,370],{"className":18852},[611],[19,18854],{"className":18855,"style":607},[271],[19,18857,18859,18862,18995],{"className":18858},[103],[19,18860],{"className":18861,"style":7849},[107],[19,18863,18865,18868,18992],{"className":18864},[155],[19,18866],{"className":18867},[177,178],[19,18869,18871],{"className":18870},[55],[19,18872,18874,18984],{"className":18873},[126,127],[19,18875,18877,18981],{"className":18876},[131],[19,18878,18880,18944,18952],{"className":18879,"style":7868},[135],[19,18881,18882,18885],{"style":3956},[19,18883],{"className":18884,"style":144},[143],[19,18886,18888,18891,18894,18897,18900,18903,18906],{"className":18887},[155],[19,18889,412],{"className":18890},[177],[19,18892,77],{"className":18893},[155,236],[19,18895],{"className":18896,"style":920},[271],[19,18898,1066],{"className":18899},[788],[19,18901],{"className":18902,"style":920},[271],[19,18904,402],{"className":18905},[155],[19,18907,18909,18912],{"className":18908},[255],[19,18910,422],{"className":18911},[255],[19,18913,18915],{"className":18914},[122],[19,18916,18918],{"className":18917},[126],[19,18919,18921],{"className":18920},[131],[19,18922,18924],{"className":18923,"style":12926},[135],[19,18925,18926,18929],{"style":12929},[19,18927],{"className":18928,"style":298},[143],[19,18930,18932],{"className":18931},[148,149,150,151],[19,18933,18935,18938,18941],{"className":18934},[155,151],[19,18936,77],{"className":18937},[155,236,151],[19,18939,1066],{"className":18940},[788,151],[19,18942,402],{"className":18943},[155,151],[19,18945,18946,18949],{"style":3968},[19,18947],{"className":18948,"style":144},[143],[19,18950],{"className":18951,"style":3975},[219],[19,18953,18954,18957],{"style":3978},[19,18955],{"className":18956,"style":144},[143],[19,18958,18960,18963,18966,18969,18972,18975,18978],{"className":18959},[155],[19,18961,412],{"className":18962},[177],[19,18964,77],{"className":18965},[155,236],[19,18967],{"className":18968,"style":920},[271],[19,18970,1066],{"className":18971},[788],[19,18973],{"className":18974,"style":920},[271],[19,18976,402],{"className":18977},[155],[19,18979,12720],{"className":18980},[255],[19,18982,242],{"className":18983},[241],[19,18985,18987],{"className":18986},[131],[19,18988,18990],{"className":18989,"style":7939},[135],[19,18991],{},[19,18993],{"className":18994},[255,178],[19,18996,18998,19001],{"className":18997},[155],[19,18999,15064],{"className":19000},[155,236],[19,19002,19004],{"className":19003},[122],[19,19005,19007],{"className":19006},[126],[19,19008,19010],{"className":19009},[131],[19,19011,19013],{"className":19012,"style":2133},[135],[19,19014,19015,19018],{"style":2136},[19,19016],{"className":19017,"style":298},[143],[19,19019,19021],{"className":19020},[148,149,150,151],[19,19022,19024,19027,19030],{"className":19023},[155,151],[19,19025,77],{"className":19026},[155,236,151],[19,19028,1066],{"className":19029},[788,151],[19,19031,402],{"className":19032},[155,151],[19,19034,19036],{"className":19035,"translate":23},[22],[19,19037,19039,19118],{"className":19038,"translate":23},[27],[19,19040,19042],{"className":19041},[31],[33,19043,19044],{"xmlns":35,"display":36},[38,19045,19046,19115],{},[41,19047,19048,19064,19084,19086,19110,19113],{},[10327,19049,19050,19056],{},[41,19051,19052,19054],{},[58,19053,10333],{},[47,19055,74],{},[41,19057,19058,19060,19062],{},[58,19059,77],{},[47,19061,10342],{},[58,19063,10346],{"mathvariant":10345},[55,19065,19066,19078],{},[18497,19067,19068,19070],{},[58,19069,18501],{},[41,19071,19072,19074,19076],{},[58,19073,77],{},[47,19075,1066],{},[51,19077,402],{},[18497,19079,19080,19082],{},[58,19081,18501],{},[58,19083,77],{},[47,19085,370],{},[55,19087,19088,19090],{},[58,19089,15064],{},[65,19091,19092,19108],{},[41,19093,19094,19096,19098,19100,19106],{},[47,19095,412],{"fence":99},[51,19097,402],{},[47,19099,1066],{},[55,19101,19102,19104],{},[51,19103,402],{},[58,19105,77],{},[47,19107,422],{"fence":99},[58,19109,77],{},[47,19111,19112],{},">",[51,19114,402],{},[91,19116,19117],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^n} > 1",[19,19119,19121,19339,19538],{"className":19120,"ariaHidden":99},[98],[19,19122,19124,19128,19181,19184,19330,19333,19336],{"className":19123},[103],[19,19125],{"className":19126,"style":19127},[107],"height:1.9436em;vertical-align:-0.836em;",[19,19129,19131],{"className":19130},[112,10645],[19,19132,19134,19173],{"className":19133},[126,127],[19,19135,19137,19170],{"className":19136},[131],[19,19138,19140,19160],{"className":19139,"style":4286},[135],[19,19141,19142,19145],{"style":15507},[19,19143],{"className":19144,"style":144},[143],[19,19146,19148],{"className":19147},[148,149,150,151],[19,19149,19151,19154,19157],{"className":19150},[155,151],[19,19152,77],{"className":19153},[155,236,151],[19,19155,10342],{"className":19156},[611,151],[19,19158,10346],{"className":19159},[155,151],[19,19161,19162,19165],{"style":10678},[19,19163],{"className":19164,"style":144},[143],[19,19166,19167],{},[19,19168,10333],{"className":19169},[112],[19,19171,242],{"className":19172},[241],[19,19174,19176],{"className":19175},[131],[19,19177,19179],{"className":19178,"style":15545},[135],[19,19180],{},[19,19182],{"className":19183,"style":272},[271],[19,19185,19187,19190,19327],{"className":19186},[155],[19,19188],{"className":19189},[177,178],[19,19191,19193],{"className":19192},[55],[19,19194,19196,19318],{"className":19195},[126,127],[19,19197,19199,19315],{"className":19198},[131],[19,19200,19202,19250,19258],{"className":19201,"style":5785},[135],[19,19203,19204,19207],{"style":3956},[19,19205],{"className":19206,"style":144},[143],[19,19208,19210],{"className":19209},[155],[19,19211,19213,19216],{"className":19212},[155],[19,19214,18501],{"className":19215},[155,236],[19,19217,19219],{"className":19218},[122],[19,19220,19222,19242],{"className":19221},[126,127],[19,19223,19225,19239],{"className":19224},[131],[19,19226,19228],{"className":19227,"style":18626},[135],[19,19229,19230,19233],{"style":18629},[19,19231],{"className":19232,"style":298},[143],[19,19234,19236],{"className":19235},[148,149,150,151],[19,19237,77],{"className":19238},[155,236,151],[19,19240,242],{"className":19241},[241],[19,19243,19245],{"className":19244},[131],[19,19246,19248],{"className":19247,"style":18648},[135],[19,19249],{},[19,19251,19252,19255],{"style":3968},[19,19253],{"className":19254,"style":144},[143],[19,19256],{"className":19257,"style":3975},[219],[19,19259,19260,19263],{"style":3978},[19,19261],{"className":19262,"style":144},[143],[19,19264,19266],{"className":19265},[155],[19,19267,19269,19272],{"className":19268},[155],[19,19270,18501],{"className":19271},[155,236],[19,19273,19275],{"className":19274},[122],[19,19276,19278,19307],{"className":19277},[126,127],[19,19279,19281,19304],{"className":19280},[131],[19,19282,19284],{"className":19283,"style":18814},[135],[19,19285,19286,19289],{"style":18629},[19,19287],{"className":19288,"style":298},[143],[19,19290,19292],{"className":19291},[148,149,150,151],[19,19293,19295,19298,19301],{"className":19294},[155,151],[19,19296,77],{"className":19297},[155,236,151],[19,19299,1066],{"className":19300},[788,151],[19,19302,402],{"className":19303},[155,151],[19,19305,242],{"className":19306},[241],[19,19308,19310],{"className":19309},[131],[19,19311,19313],{"className":19312,"style":18844},[135],[19,19314],{},[19,19316,242],{"className":19317},[241],[19,19319,19321],{"className":19320},[131],[19,19322,19325],{"className":19323,"style":19324},[135],"height:0.836em;",[19,19326],{},[19,19328],{"className":19329},[255,178],[19,19331],{"className":19332,"style":607},[271],[19,19334,370],{"className":19335},[611],[19,19337],{"className":19338,"style":607},[271],[19,19340,19342,19346,19529,19532,19535],{"className":19341},[103],[19,19343],{"className":19344,"style":19345},[107],"height:2.2519em;vertical-align:-1.1443em;",[19,19347,19349,19352,19526],{"className":19348},[155],[19,19350],{"className":19351},[177,178],[19,19353,19355],{"className":19354},[55],[19,19356,19358,19517],{"className":19357},[126,127],[19,19359,19361,19514],{"className":19360},[131],[19,19362,19364,19495,19503],{"className":19363,"style":5785},[135],[19,19365,19367,19370],{"style":19366},"top:-2.2057em;",[19,19368],{"className":19369,"style":144},[143],[19,19371,19373],{"className":19372},[155],[19,19374,19376,19472],{"className":19375},[1353],[19,19377,19379,19385,19388,19391,19394,19397,19466],{"className":19378},[1353],[19,19380,19382],{"className":19381,"style":1358},[177,1357],[19,19383,412],{"className":19384},[1362,203],[19,19386,402],{"className":19387},[155],[19,19389],{"className":19390,"style":920},[271],[19,19392,1066],{"className":19393},[788],[19,19395],{"className":19396,"style":920},[271],[19,19398,19400,19403,19463],{"className":19399},[155],[19,19401],{"className":19402},[177,178],[19,19404,19406],{"className":19405},[55],[19,19407,19409,19455],{"className":19408},[126,127],[19,19410,19412,19452],{"className":19411},[131],[19,19413,19416,19430,19438],{"className":19414,"style":19415},[135],"height:0.8451em;",[19,19417,19418,19421],{"style":8764},[19,19419],{"className":19420,"style":144},[143],[19,19422,19424],{"className":19423},[148,149,150,151],[19,19425,19427],{"className":19426},[155,151],[19,19428,77],{"className":19429},[155,236,151],[19,19431,19432,19435],{"style":3968},[19,19433],{"className":19434,"style":144},[143],[19,19436],{"className":19437,"style":3975},[219],[19,19439,19440,19443],{"style":8787},[19,19441],{"className":19442,"style":144},[143],[19,19444,19446],{"className":19445},[148,149,150,151],[19,19447,19449],{"className":19448},[155,151],[19,19450,402],{"className":19451},[155,151],[19,19453,242],{"className":19454},[241],[19,19456,19458],{"className":19457},[131],[19,19459,19461],{"className":19460,"style":8809},[135],[19,19462],{},[19,19464],{"className":19465},[255,178],[19,19467,19469],{"className":19468,"style":1358},[255,1357],[19,19470,422],{"className":19471},[1362,203],[19,19473,19475],{"className":19474},[122],[19,19476,19478],{"className":19477},[126],[19,19479,19481],{"className":19480},[131],[19,19482,19484],{"className":19483,"style":15397},[135],[19,19485,19486,19489],{"style":15400},[19,19487],{"className":19488,"style":298},[143],[19,19490,19492],{"className":19491},[148,149,150,151],[19,19493,77],{"className":19494},[155,236,151],[19,19496,19497,19500],{"style":3968},[19,19498],{"className":19499,"style":144},[143],[19,19501],{"className":19502,"style":3975},[219],[19,19504,19505,19508],{"style":3978},[19,19506],{"className":19507,"style":144},[143],[19,19509,19511],{"className":19510},[155],[19,19512,15064],{"className":19513},[155,236],[19,19515,242],{"className":19516},[241],[19,19518,19520],{"className":19519},[131],[19,19521,19524],{"className":19522,"style":19523},[135],"height:1.1443em;",[19,19525],{},[19,19527],{"className":19528},[255,178],[19,19530],{"className":19531,"style":607},[271],[19,19533,19112],{"className":19534},[611],[19,19536],{"className":19537,"style":607},[271],[19,19539,19541,19544],{"className":19540},[103],[19,19542],{"className":19543,"style":5939},[107],[19,19545,402],{"className":19546},[155],[15,19548,19549,19550,19578,19579,19607,19608,19678],{},"Although the limit of the expression is ",[19,19551,19553,19566],{"className":19552,"translate":23},[27],[19,19554,19556],{"className":19555},[31],[33,19557,19558],{"xmlns":35},[38,19559,19560,19564],{},[41,19561,19562],{},[58,19563,15064],{},[91,19565,15064],{"encoding":93},[19,19567,19569],{"className":19568,"ariaHidden":99},[98],[19,19570,19572,19575],{"className":19571},[103],[19,19573],{"className":19574,"style":4230},[107],[19,19576,15064],{"className":19577},[155,236],", it is always approaching ",[19,19580,19582,19595],{"className":19581,"translate":23},[27],[19,19583,19585],{"className":19584},[31],[33,19586,19587],{"xmlns":35},[38,19588,19589,19593],{},[41,19590,19591],{},[58,19592,15064],{},[91,19594,15064],{"encoding":93},[19,19596,19598],{"className":19597,"ariaHidden":99},[98],[19,19599,19601,19604],{"className":19600},[103],[19,19602],{"className":19603,"style":4230},[107],[19,19605,15064],{"className":19606},[155,236]," from below, so it is still greater than 1, indicating that ",[19,19609,19611,19629],{"className":19610,"translate":23},[27],[19,19612,19614],{"className":19613},[31],[33,19615,19616],{"xmlns":35},[38,19617,19618,19626],{},[41,19619,19620],{},[18497,19621,19622,19624],{},[58,19623,18501],{},[58,19625,77],{},[91,19627,19628],{"encoding":93},"a_n",[19,19630,19632],{"className":19631,"ariaHidden":99},[98],[19,19633,19635,19638],{"className":19634},[103],[19,19636],{"className":19637,"style":18607},[107],[19,19639,19641,19644],{"className":19640},[155],[19,19642,18501],{"className":19643},[155,236],[19,19645,19647],{"className":19646},[122],[19,19648,19650,19670],{"className":19649},[126,127],[19,19651,19653,19667],{"className":19652},[131],[19,19654,19656],{"className":19655,"style":18626},[135],[19,19657,19658,19661],{"style":18629},[19,19659],{"className":19660,"style":298},[143],[19,19662,19664],{"className":19663},[148,149,150,151],[19,19665,77],{"className":19666},[155,236,151],[19,19668,242],{"className":19669},[241],[19,19671,19673],{"className":19672},[131],[19,19674,19676],{"className":19675,"style":18648},[135],[19,19677],{}," is divergent.",[15,19680,19681],{},"Mathematicians have attempted various approaches and found:",[19,19683,19685],{"className":19684,"translate":23},[22],[19,19686,19688,19738],{"className":19687,"translate":23},[27],[19,19689,19691],{"className":19690},[31],[33,19692,19693],{"xmlns":35,"display":36},[38,19694,19695,19735],{},[41,19696,19697,19703,19705,19729],{},[18497,19698,19699,19701],{},[58,19700,18501],{},[58,19702,77],{},[47,19704,370],{},[55,19706,19707,19713],{},[41,19708,19709,19711],{},[58,19710,77],{},[47,19712,7425],{"stretchy":411},[65,19714,19715,19717],{},[58,19716,77],{},[41,19718,19719,19721,19723],{},[58,19720,77],{},[47,19722,1066],{},[55,19724,19725,19727],{},[51,19726,402],{},[51,19728,63],{},[65,19730,19731,19733],{},[58,19732,15064],{},[58,19734,77],{},[91,19736,19737],{"encoding":93},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n",[19,19739,19741,19796],{"className":19740,"ariaHidden":99},[98],[19,19742,19744,19747,19787,19790,19793],{"className":19743},[103],[19,19745],{"className":19746,"style":18607},[107],[19,19748,19750,19753],{"className":19749},[155],[19,19751,18501],{"className":19752},[155,236],[19,19754,19756],{"className":19755},[122],[19,19757,19759,19779],{"className":19758},[126,127],[19,19760,19762,19776],{"className":19761},[131],[19,19763,19765],{"className":19764,"style":18626},[135],[19,19766,19767,19770],{"style":18629},[19,19768],{"className":19769,"style":298},[143],[19,19771,19773],{"className":19772},[148,149,150,151],[19,19774,77],{"className":19775},[155,236,151],[19,19777,242],{"className":19778},[241],[19,19780,19782],{"className":19781},[131],[19,19783,19785],{"className":19784,"style":18648},[135],[19,19786],{},[19,19788],{"className":19789,"style":607},[271],[19,19791,370],{"className":19792},[611],[19,19794],{"className":19795,"style":607},[271],[19,19797,19799,19803,19972],{"className":19798},[103],[19,19800],{"className":19801,"style":19802},[107],"height:2.201em;vertical-align:-0.8296em;",[19,19804,19806,19809,19969],{"className":19805},[155],[19,19807],{"className":19808},[177,178],[19,19810,19812],{"className":19811},[55],[19,19813,19815,19960],{"className":19814},[126,127],[19,19816,19818,19957],{"className":19817},[131],[19,19819,19821,19935,19943],{"className":19820,"style":4771},[135],[19,19822,19824,19827],{"style":19823},"top:-2.1704em;",[19,19825],{"className":19826,"style":144},[143],[19,19828,19830],{"className":19829},[155],[19,19831,19833,19836],{"className":19832},[155],[19,19834,77],{"className":19835},[155,236],[19,19837,19839],{"className":19838},[122],[19,19840,19842],{"className":19841},[126],[19,19843,19845],{"className":19844},[131],[19,19846,19849],{"className":19847,"style":19848},[135],"height:0.9396em;",[19,19850,19852,19855],{"style":19851},"top:-3.3486em;margin-right:0.05em;",[19,19853],{"className":19854,"style":144},[143],[19,19856,19858],{"className":19857},[148,149,150,151],[19,19859,19861,19864,19867],{"className":19860},[155,151],[19,19862,77],{"className":19863},[155,236,151],[19,19865,1066],{"className":19866},[788,151],[19,19868,19870,19873,19932],{"className":19869},[155,151],[19,19871],{"className":19872},[177,178,148,179,180],[19,19874,19876],{"className":19875},[55],[19,19877,19879,19924],{"className":19878},[126,127],[19,19880,19882,19921],{"className":19881},[131],[19,19883,19885,19899,19907],{"className":19884,"style":15704},[135],[19,19886,19887,19890],{"style":196},[19,19888],{"className":19889,"style":144},[143],[19,19891,19893],{"className":19892},[148,179,203,151],[19,19894,19896],{"className":19895},[155,151],[19,19897,63],{"className":19898},[155,151],[19,19900,19901,19904],{"style":212},[19,19902],{"className":19903,"style":144},[143],[19,19905],{"className":19906,"style":220},[219,151],[19,19908,19909,19912],{"style":223},[19,19910],{"className":19911,"style":144},[143],[19,19913,19915],{"className":19914},[148,179,203,151],[19,19916,19918],{"className":19917},[155,151],[19,19919,402],{"className":19920},[155,151],[19,19922,242],{"className":19923},[241],[19,19925,19927],{"className":19926},[131],[19,19928,19930],{"className":19929,"style":249},[135],[19,19931],{},[19,19933],{"className":19934},[255,178,148,179,180],[19,19936,19937,19940],{"style":3968},[19,19938],{"className":19939,"style":144},[143],[19,19941],{"className":19942,"style":3975},[219],[19,19944,19945,19948],{"style":3978},[19,19946],{"className":19947,"style":144},[143],[19,19949,19951,19954],{"className":19950},[155],[19,19952,77],{"className":19953},[155,236],[19,19955,7425],{"className":19956},[255],[19,19958,242],{"className":19959},[241],[19,19961,19963],{"className":19962},[131],[19,19964,19967],{"className":19965,"style":19966},[135],"height:0.8296em;",[19,19968],{},[19,19970],{"className":19971},[255,178],[19,19973,19975,19978],{"className":19974},[155],[19,19976,15064],{"className":19977},[155,236],[19,19979,19981],{"className":19980},[122],[19,19982,19984],{"className":19983},[126],[19,19985,19987],{"className":19986},[131],[19,19988,19990],{"className":19989,"style":18443},[135],[19,19991,19992,19995],{"style":2136},[19,19993],{"className":19994,"style":298},[143],[19,19996,19998],{"className":19997},[148,149,150,151],[19,19999,77],{"className":20000},[155,236,151],[15,20002,20003,20004,20073,20074,20077],{},"To prove that ",[19,20005,20007,20024],{"className":20006,"translate":23},[27],[19,20008,20010],{"className":20009},[31],[33,20011,20012],{"xmlns":35},[38,20013,20014,20022],{},[41,20015,20016],{},[18497,20017,20018,20020],{},[58,20019,18501],{},[58,20021,77],{},[91,20023,19628],{"encoding":93},[19,20025,20027],{"className":20026,"ariaHidden":99},[98],[19,20028,20030,20033],{"className":20029},[103],[19,20031],{"className":20032,"style":18607},[107],[19,20034,20036,20039],{"className":20035},[155],[19,20037,18501],{"className":20038},[155,236],[19,20040,20042],{"className":20041},[122],[19,20043,20045,20065],{"className":20044},[126,127],[19,20046,20048,20062],{"className":20047},[131],[19,20049,20051],{"className":20050,"style":18626},[135],[19,20052,20053,20056],{"style":18629},[19,20054],{"className":20055,"style":298},[143],[19,20057,20059],{"className":20058},[148,149,150,151],[19,20060,77],{"className":20061},[155,236,151],[19,20063,242],{"className":20064},[241],[19,20066,20068],{"className":20067},[131],[19,20069,20071],{"className":20070,"style":18648},[135],[19,20072],{}," converges, use the ",[4202,20075,20076],{},"ratio test"," 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\\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}}",[19,20866,20868,21084],{"className":20867,"ariaHidden":99},[98],[19,20869,20871,20874,20927,20930,21075,21078,21081],{"className":20870},[103],[19,20872],{"className":20873,"style":19127},[107],[19,20875,20877],{"className":20876},[112,10645],[19,20878,20880,20919],{"className":20879},[126,127],[19,20881,20883,20916],{"className":20882},[131],[19,20884,20886,20906],{"className":20885,"style":4286},[135],[19,20887,20888,20891],{"style":15507},[19,20889],{"className":20890,"style":144},[143],[19,20892,20894],{"className":20893},[148,149,150,151],[19,20895,20897,20900,20903],{"className":20896},[155,151],[19,20898,77],{"className":20899},[155,236,151],[19,20901,10342],{"className":20902},[611,151],[19,20904,10346],{"className":20905},[155,151],[19,20907,20908,20911],{"style":10678},[19,20909],{"className":20910,"style":144},[143],[19,20912,20913],{},[19,20914,10333],{"className":20915},[112],[19,20917,242],{"className":20918},[241],[19,20920,20922],{"className":20921},[131],[19,20923,20925],{"className":20924,"style":15545},[135],[19,20926],{},[19,20928],{"className":20929,"style":272},[271],[19,20931,20933,20936,21072],{"className":20932},[155],[19,20934],{"className":20935},[177,178],[19,20937,20939],{"className":20938},[55],[19,20940,20942,21064],{"className":20941},[126,127],[19,20943,20945,21061],{"className":20944},[131],[19,20946,20948,20996,21004],{"className":20947,"style":5785},[135],[19,20949,20950,20953],{"style":3956},[19,20951],{"className":20952,"style":144},[143],[19,20954,20956],{"className":20955},[155],[19,20957,20959,20962],{"className":20958},[155],[19,20960,18501],{"className":20961},[155,236],[19,20963,20965],{"className":20964},[122],[19,20966,20968,20988],{"className":20967},[126,127],[19,20969,20971,20985],{"className":20970},[131],[19,20972,20974],{"className":20973,"style":18626},[135],[19,20975,20976,20979],{"style":18629},[19,20977],{"className":20978,"style":298},[143],[19,20980,20982],{"className":20981},[148,149,150,151],[19,20983,77],{"className":20984},[155,236,151],[19,20986,242],{"className":20987},[241],[19,20989,20991],{"className":20990},[131],[19,20992,20994],{"className":20993,"style":18648},[135],[19,20995],{},[19,20997,20998,21001],{"style":3968},[19,20999],{"className":21000,"style":144},[143],[19,21002],{"className":21003,"style":3975},[219],[19,21005,21006,21009],{"style":3978},[19,21007],{"className":21008,"style":144},[143],[19,21010,21012],{"className":21011},[155],[19,21013,21015,21018],{"className":21014},[155],[19,21016,18501],{"className":21017},[155,236],[19,21019,21021],{"className":21020},[122],[19,21022,21024,21053],{"className":21023},[126,127],[19,21025,21027,21050],{"className":21026},[131],[19,21028,21030],{"className":21029,"style":18814},[135],[19,21031,21032,21035],{"style":18629},[19,21033],{"className":21034,"style":298},[143],[19,21036,21038],{"className":21037},[148,149,150,151],[19,21039,21041,21044,21047],{"className":21040},[155,151],[19,21042,77],{"className":21043},[155,236,151],[19,21045,1066],{"className":21046},[788,151],[19,21048,402],{"className":21049},[155,151],[19,21051,242],{"className":21052},[241],[19,21054,21056],{"className":21055},[131],[19,21057,21059],{"className":21058,"style":18844},[135],[19,21060],{},[19,21062,242],{"className":21063},[241],[19,21065,21067],{"className":21066},[131],[19,21068,21070],{"className":21069,"style":19324},[135],[19,21071],{},[19,21073],{"className":21074},[255,178],[19,21076],{"className":21077,"style":607},[271],[19,21079,370],{"className":21080},[611],[19,21082],{"className":21083,"style":607},[271],[19,21085,21087,21091],{"className":21086},[103],[19,21088],{"className":21089,"style":21090},[107],"height:2.5415em;vertical-align:-1.4339em;",[19,21092,21094,21097,21348],{"className":21093},[155],[19,21095],{"className":21096},[177,178],[19,21098,21100],{"className":21099},[55],[19,21101,21103,21339],{"className":21102},[126,127],[19,21104,21106,21336],{"className":21105},[131],[19,21107,21109,21315,21324],{"className":21108,"style":5785},[135],[19,21110,21112,21116],{"style":21111},"top:-2.11em;",[19,21113],{"className":21114,"style":21115},[143],"height:3.1939em;",[19,21117,21119],{"className":21118},[155],[19,21120,21122,21217],{"className":21121},[1353],[19,21123,21125,21131,21134,21137,21140,21143,21211],{"className":21124},[1353],[19,21126,21128],{"className":21127,"style":1358},[177,1357],[19,21129,412],{"className":21130},[1362,203],[19,21132,402],{"className":21133},[155],[19,21135],{"className":21136,"style":920},[271],[19,21138,1066],{"className":21139},[788],[19,21141],{"className":21142,"style":920},[271],[19,21144,21146,21149,21208],{"className":21145},[155],[19,21147],{"className":21148},[177,178],[19,21150,21152],{"className":21151},[55],[19,21153,21155,21200],{"className":21154},[126,127],[19,21156,21158,21197],{"className":21157},[131],[19,21159,21161,21175,21183],{"className":21160,"style":19415},[135],[19,21162,21163,21166],{"style":8764},[19,21164],{"className":21165,"style":144},[143],[19,21167,21169],{"className":21168},[148,149,150,151],[19,21170,21172],{"className":21171},[155,151],[19,21173,77],{"className":21174},[155,236,151],[19,21176,21177,21180],{"style":3968},[19,21178],{"className":21179,"style":144},[143],[19,21181],{"className":21182,"style":3975},[219],[19,21184,21185,21188],{"style":8787},[19,21186],{"className":21187,"style":144},[143],[19,21189,21191],{"className":21190},[148,149,150,151],[19,21192,21194],{"className":21193},[155,151],[19,21195,402],{"className":21196},[155,151],[19,21198,242],{"className":21199},[241],[19,21201,21203],{"className":21202},[131],[19,21204,21206],{"className":21205,"style":8809},[135],[19,21207],{},[19,21209],{"className":21210},[255,178],[19,21212,21214],{"className":21213,"style":1358},[255,1357],[19,21215,422],{"className":21216},[1362,203],[19,21218,21220],{"className":21219},[122],[19,21221,21223],{"className":21222},[126],[19,21224,21226],{"className":21225},[131],[19,21227,21230],{"className":21228,"style":21229},[135],"height:1.1939em;",[19,21231,21232,21235],{"style":17908},[19,21233],{"className":21234,"style":144},[143],[19,21236,21238],{"className":21237},[148,149,150,151],[19,21239,21241,21244,21247],{"className":21240},[155,151],[19,21242,77],{"className":21243},[155,236,151],[19,21245,1066],{"className":21246},[788,151],[19,21248,21250,21253,21312],{"className":21249},[155,151],[19,21251],{"className":21252},[177,178,148,179,180],[19,21254,21256],{"className":21255},[55],[19,21257,21259,21304],{"className":21258},[126,127],[19,21260,21262,21301],{"className":21261},[131],[19,21263,21265,21279,21287],{"className":21264,"style":15704},[135],[19,21266,21267,21270],{"style":196},[19,21268],{"className":21269,"style":144},[143],[19,21271,21273],{"className":21272},[148,179,203,151],[19,21274,21276],{"className":21275},[155,151],[19,21277,63],{"className":21278},[155,151],[19,21280,21281,21284],{"style":212},[19,21282],{"className":21283,"style":144},[143],[19,21285],{"className":21286,"style":220},[219,151],[19,21288,21289,21292],{"style":223},[19,21290],{"className":21291,"style":144},[143],[19,21293,21295],{"className":21294},[148,179,203,151],[19,21296,21298],{"className":21297},[155,151],[19,21299,402],{"className":21300},[155,151],[19,21302,242],{"className":21303},[241],[19,21305,21307],{"className":21306},[131],[19,21308,21310],{"className":21309,"style":249},[135],[19,21311],{},[19,21313],{"className":21314},[255,178,148,179,180],[19,21316,21318,21321],{"style":21317},"top:-3.4239em;",[19,21319],{"className":21320,"style":21115},[143],[19,21322],{"className":21323,"style":3975},[219],[19,21325,21327,21330],{"style":21326},"top:-3.8709em;",[19,21328],{"className":21329,"style":21115},[143],[19,21331,21333],{"className":21332},[155],[19,21334,15064],{"className":21335},[155,236],[19,21337,242],{"className":21338},[241],[19,21340,21342],{"className":21341},[131],[19,21343,21346],{"className":21344,"style":21345},[135],"height:1.4339em;",[19,21347],{},[19,21349],{"className":21350},[255,178],[15,21352,21353],{},"Compare this with:",[19,21355,21357],{"className":21356,"translate":23},[22],[19,21358,21360,21444],{"className":21359,"translate":23},[27],[19,21361,21363],{"className":21362},[31],[33,21364,21365],{"xmlns":35,"display":36},[38,21366,21367,21441],{},[41,21368,21369,21399,21401],{},[65,21370,21371,21387],{},[41,21372,21373,21375,21377,21379,21385],{},[47,21374,412],{"fence":99},[51,21376,402],{},[47,21378,1066],{},[55,21380,21381,21383],{},[51,21382,402],{},[58,21384,77],{},[47,21386,422],{"fence":99},[41,21388,21389,21391,21393],{},[58,21390,77],{},[47,21392,1066],{},[55,21394,21395,21397],{},[51,21396,402],{},[51,21398,63],{},[47,21400,370],{},[65,21402,21403,21405],{},[58,21404,15064],{},[41,21406,21407,21409,21411,21413,21419,21421,21423,21425],{},[47,21408,412],{"stretchy":411},[58,21410,77],{},[47,21412,1066],{},[55,21414,21415,21417],{},[51,21416,402],{},[51,21418,63],{},[47,21420,422],{"stretchy":411},[58,21422,15927],{},[47,21424,74],{},[41,21426,21427,21429,21431,21433,21439],{},[47,21428,412],{"fence":99},[51,21430,402],{},[47,21432,1066],{},[55,21434,21435,21437],{},[51,21436,402],{},[58,21438,77],{},[47,21440,422],{"fence":99},[91,21442,21443],{"encoding":93},"\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}} = e^{(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)}",[19,21445,21447,21651],{"className":21446,"ariaHidden":99},[98],[19,21448,21450,21453,21642,21645,21648],{"className":21449},[103],[19,21451],{"className":21452,"style":15492},[107],[19,21454,21456,21545],{"className":21455},[1353],[19,21457,21459,21465,21468,21471,21474,21477,21539],{"className":21458},[1353],[19,21460,21462],{"className":21461,"style":1358},[177,1357],[19,21463,412],{"className":21464},[1362,150],[19,21466,402],{"className":21467},[155],[19,21469],{"className":21470,"style":920},[271],[19,21472,1066],{"className":21473},[788],[19,21475],{"className":21476,"style":920},[271],[19,21478,21480,21483,21536],{"className":21479},[155],[19,21481],{"className":21482},[177,178],[19,21484,21486],{"className":21485},[55],[19,21487,21489,21528],{"className":21488},[126,127],[19,21490,21492,21525],{"className":21491},[131],[19,21493,21495,21506,21514],{"className":21494,"style":3953},[135],[19,21496,21497,21500],{"style":3956},[19,21498],{"className":21499,"style":144},[143],[19,21501,21503],{"className":21502},[155],[19,21504,77],{"className":21505},[155,236],[19,21507,21508,21511],{"style":3968},[19,21509],{"className":21510,"style":144},[143],[19,21512],{"className":21513,"style":3975},[219],[19,21515,21516,21519],{"style":3978},[19,21517],{"className":21518,"style":144},[143],[19,21520,21522],{"className":21521},[155],[19,21523,402],{"className":21524},[155],[19,21526,242],{"className":21527},[241],[19,21529,21531],{"className":21530},[131],[19,21532,21534],{"className":21533,"style":4009},[135],[19,21535],{},[19,21537],{"className":21538},[255,178],[19,21540,21542],{"className":21541,"style":1358},[255,1357],[19,21543,422],{"className":21544},[1362,150],[19,21546,21548],{"className":21547},[122],[19,21549,21551],{"className":21550},[126],[19,21552,21554],{"className":21553},[131],[19,21555,21557],{"className":21556,"style":15673},[135],[19,21558,21559,21562],{"style":15676},[19,21560],{"className":21561,"style":144},[143],[19,21563,21565],{"className":21564},[148,149,150,151],[19,21566,21568,21571,21574],{"className":21567},[155,151],[19,21569,77],{"className":21570},[155,236,151],[19,21572,1066],{"className":21573},[788,151],[19,21575,21577,21580,21639],{"className":21576},[155,151],[19,21578],{"className":21579},[177,178,148,179,180],[19,21581,21583],{"className":21582},[55],[19,21584,21586,21631],{"className":21585},[126,127],[19,21587,21589,21628],{"className":21588},[131],[19,21590,21592,21606,21614],{"className":21591,"style":15704},[135],[19,21593,21594,21597],{"style":196},[19,21595],{"className":21596,"style":144},[143],[19,21598,21600],{"className":21599},[148,179,203,151],[19,21601,21603],{"className":21602},[155,151],[19,21604,63],{"className":21605},[155,151],[19,21607,21608,21611],{"style":212},[19,21609],{"className":21610,"style":144},[143],[19,21612],{"className":21613,"style":220},[219,151],[19,21615,21616,21619],{"style":223},[19,21617],{"className":21618,"style":144},[143],[19,21620,21622],{"className":21621},[148,179,203,151],[19,21623,21625],{"className":21624},[155,151],[19,21626,402],{"className":21627},[155,151],[19,21629,242],{"className":21630},[241],[19,21632,21634],{"className":21633},[131],[19,21635,21637],{"className":21636,"style":249},[135],[19,21638],{},[19,21640],{"className":21641},[255,178,148,179,180],[19,21643],{"className":21644,"style":607},[271],[19,21646,370],{"className":21647},[611],[19,21649],{"className":21650,"style":607},[271],[19,21652,21654,21657],{"className":21653},[103],[19,21655],{"className":21656,"style":16308},[107],[19,21658,21660,21663],{"className":21659},[155],[19,21661,15064],{"className":21662},[155,236],[19,21664,21666],{"className":21665},[122],[19,21667,21669],{"className":21668},[126],[19,21670,21672],{"className":21671},[131],[19,21673,21675],{"className":21674,"style":16308},[135],[19,21676,21677,21680],{"style":16311},[19,21678],{"className":21679,"style":144},[143],[19,21681,21683],{"className":21682},[148,149,150,151],[19,21684,21686,21689,21692,21695,21763,21766,21769,21778],{"className":21685},[155,151],[19,21687,412],{"className":21688},[177,151],[19,21690,77],{"className":21691},[155,236,151],[19,21693,1066],{"className":21694},[788,151],[19,21696,21698,21701,21760],{"className":21697},[155,151],[19,21699],{"className":21700},[177,178,148,179,180],[19,21702,21704],{"className":21703},[55],[19,21705,21707,21752],{"className":21706},[126,127],[19,21708,21710,21749],{"className":21709},[131],[19,21711,21713,21727,21735],{"className":21712,"style":15704},[135],[19,21714,21715,21718],{"style":196},[19,21716],{"className":21717,"style":144},[143],[19,21719,21721],{"className":21720},[148,179,203,151],[19,21722,21724],{"className":21723},[155,151],[19,21725,63],{"className":21726},[155,151],[19,21728,21729,21732],{"style":212},[19,21730],{"className":21731,"style":144},[143],[19,21733],{"className":21734,"style":220},[219,151],[19,21736,21737,21740],{"style":223},[19,21738],{"className":21739,"style":144},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",[19,21871,21873,21886],{"className":21872,"translate":23},[27],[19,21874,21876],{"className":21875},[31],[33,21877,21878],{"xmlns":35},[38,21879,21880,21884],{},[41,21881,21882],{},[58,21883,15064],{},[91,21885,15064],{"encoding":93},[19,21887,21889],{"className":21888,"ariaHidden":99},[98],[19,21890,21892,21895],{"className":21891},[103],[19,21893],{"className":21894,"style":4230},[107],[19,21896,15064],{"className":21897},[155,236],[15,21899,21900],{},"Thus, compare:",[19,21902,21904],{"className":21903,"translate":23},[22],[19,21905,21907,21953],{"className":21906,"translate":23},[27],[19,21908,21910],{"className":21909},[31],[33,21911,21912],{"xmlns":35,"display":36},[38,21913,21914,21950],{},[41,21915,21916,21918,21920,21922,21928,21930,21932,21934],{},[47,21917,412],{"stretchy":411},[58,21919,77],{},[47,21921,1066],{},[55,21923,21924,21926],{},[51,21925,402],{},[51,21927,63],{},[47,21929,422],{"stretchy":411},[58,21931,15927],{},[47,21933,74],{},[41,21935,21936,21938,21940,21942,21948],{},[47,21937,412],{"fence":99},[51,21939,402],{},[47,21941,1066],{},[55,21943,21944,21946],{},[51,21945,402],{},[58,21947,77],{},[47,21949,422],{"fence":99},[91,21951,21952],{"encoding":93},"(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)",[19,21954,21956,21977],{"className":21955,"ariaHidden":99},[98],[19,21957,21959,21962,21965,21968,21971,21974],{"className":21958},[103],[19,21960],{"className":21961,"style":2167},[107],[19,21963,412],{"className":21964},[177],[19,21966,77],{"className":21967},[155,236],[19,21969],{"className":21970,"style":920},[271],[19,21972,1066],{"className":21973},[788],[19,21975],{"className":21976,"style":920},[271],[19,21978,21980,21983,22045,22048,22051,22054,22057],{"className":21979},[103],[19,21981],{"className":21982,"style":10641},[107],[19,21984,21986,21989,22042],{"className":21985},[155],[19,21987],{"className":21988},[177,178],[19,21990,21992],{"className":21991},[55],[19,21993,21995,22034],{"className":21994},[126,127],[19,21996,21998,22031],{"className":21997},[131],[19,21999,22001,22012,22020],{"className":22000,"style":3953},[135],[19,22002,22003,22006],{"style":3956},[19,22004],{"className":22005,"style":144},[143],[19,22007,22009],{"className":22008},[155],[19,22010,63],{"className":22011},[155],[19,22013,22014,22017],{"style":3968},[19,22015],{"className":22016,"style":144},[143],[19,22018],{"className":22019,"style":3975},[219],[19,22021,22022,22025],{"style":3978},[19,22023],{"className":22024,"style":144},[143],[19,22026,22028],{"className":22027},[155],[19,22029,402],{"className":22030},[155],[19,22032,242],{"className":22033},[241],[19,22035,22037],{"className":22036},[131],[19,22038,22040],{"className":22039,"style":4009},[135],[19,22041],{},[19,22043],{"className":22044},[255,178],[19,22046,422],{"className":22047},[255],[19,22049],{"className":22050,"style":272},[271],[19,22052,15927],{"className":22053},[112],[19,22055],{"className":22056,"style":272},[271],[19,22058,22060,22066,22069,22072,22075,22078,22140],{"className":22059},[1353],[19,22061,22063],{"className":22062,"style":1358},[177,1357],[19,22064,412],{"className":22065},[1362,150],[19,22067,402],{"className":22068},[155],[19,22070],{"className":22071,"style":920},[271],[19,22073,1066],{"className":22074},[788],[19,22076],{"className":22077,"style":920},[271],[19,22079,22081,22084,22137],{"className":22080},[155],[19,22082],{"className":22083},[177,178],[19,22085,22087],{"className":22086},[55],[19,22088,22090,22129],{"className":22089},[126,127],[19,22091,22093,22126],{"className":22092},[131],[19,22094,22096,22107,22115],{"className":22095,"style":3953},[135],[19,22097,22098,22101],{"style":3956},[19,22099],{"className":22100,"style":144},[143],[19,22102,22104],{"className":22103},[155],[19,22105,77],{"className":22106},[155,236],[19,22108,22109,22112],{"style":3968},[19,22110],{"className":22111,"style":144},[143],[19,22113],{"className":22114,"style":3975},[219],[19,22116,22117,22120],{"style":3978},[19,22118],{"className":22119,"style":144},[143],[19,22121,22123],{"className":22122},[155],[19,22124,402],{"className":22125},[155],[19,22127,242],{"className":22128},[241],[19,22130,22132],{"className":22131},[131],[19,22133,22135],{"className":22134,"style":4009},[135],[19,22136],{},[19,22138],{"className":22139},[255,178],[19,22141,22143],{"className":22142,"style":1358},[255,1357],[19,22144,422],{"className":22145},[1362,150],[15,22147,22148],{},"with 1.",[15,22150,22151,22152,22273,22274,4294],{},"Specifically, compare ",[19,22153,22155,22183],{"className":22154,"translate":23},[27],[19,22156,22158],{"className":22157},[31],[33,22159,22160],{"xmlns":35},[38,22161,22162,22180],{},[41,22163,22164,22166,22168],{},[58,22165,15927],{},[47,22167,74],{},[55,22169,22170,22178],{},[41,22171,22172,22174,22176],{},[58,22173,77],{},[47,22175,1066],{},[51,22177,402],{},[58,22179,77],{},[91,22181,22182],{"encoding":93},"\\ln \\frac{n+1}{n}",[19,22184,22186],{"className":22185,"ariaHidden":99},[98],[19,22187,22189,22193,22196,22199],{"className":22188},[103],[19,22190],{"className":22191,"style":22192},[107],"height:1.1901em;vertical-align:-0.345em;",[19,22194,15927],{"className":22195},[112],[19,22197],{"className":22198,"style":272},[271],[19,22200,22202,22205,22270],{"className":22201},[155],[19,22203],{"className":22204},[177,178],[19,22206,22208],{"className":22207},[55],[19,22209,22211,22262],{"className":22210},[126,127],[19,22212,22214,22259],{"className":22213},[131],[19,22215,22217,22231,22239],{"className":22216,"style":19415},[135],[19,22218,22219,22222],{"style":8764},[19,22220],{"className":22221,"style":144},[143],[19,22223,22225],{"className":22224},[148,149,150,151],[19,22226,22228],{"className":22227},[155,151],[19,22229,77],{"className":22230},[155,236,151],[19,22232,22233,22236],{"style":3968},[19,22234],{"className":22235,"style":144},[143],[19,22237],{"className":22238,"style":3975},[219],[19,22240,22241,22244],{"style":8787},[19,22242],{"className":22243,"style":144},[143],[19,22245,22247],{"className":22246},[148,149,150,151],[19,22248,22250,22253,22256],{"className":22249},[155,151],[19,22251,77],{"className":22252},[155,236,151],[19,22254,1066],{"className":22255},[788,151],[19,22257,402],{"className":22258},[155,151],[19,22260,242],{"className":22261},[241],[19,22263,22265],{"className":22264},[131],[19,22266,22268],{"className":22267,"style":8809},[135],[19,22269],{},[19,22271],{"className":22272},[255,178]," with ",[19,22275,22277,22305],{"className":22276,"translate":23},[27],[19,22278,22280],{"className":22279},[31],[33,22281,22282],{"xmlns":35},[38,22283,22284,22302],{},[41,22285,22286],{},[55,22287,22288,22290],{},[51,22289,402],{},[41,22291,22292,22294,22296],{},[58,22293,77],{},[47,22295,1066],{},[55,22297,22298,22300],{},[51,22299,402],{},[51,22301,63],{},[91,22303,22304],{"encoding":93},"\\frac{1}{n + \\frac{1}{2}}",[19,22306,22308],{"className":22307,"ariaHidden":99},[98],[19,22309,22311,22315],{"className":22310},[103],[19,22312],{"className":22313,"style":22314},[107],"height:1.4869em;vertical-align:-0.6418em;",[19,22316,22318,22321,22453],{"className":22317},[155],[19,22319],{"className":22320},[177,178],[19,22322,22324],{"className":22323},[55],[19,22325,22327,22444],{"className":22326},[126,127],[19,22328,22330,22441],{"className":22329},[131],[19,22331,22333,22419,22427],{"className":22332,"style":19415},[135],[19,22334,22336,22339],{"style":22335},"top:-2.599em;",[19,22337],{"className":22338,"style":144},[143],[19,22340,22342],{"className":22341},[148,149,150,151],[19,22343,22345,22348,22351],{"className":22344},[155,151],[19,22346,77],{"className":22347},[155,236,151],[19,22349,1066],{"className":22350},[788,151],[19,22352,22354,22357,22416],{"className":22353},[155,151],[19,22355],{"className":22356},[177,178,148,179,180],[19,22358,22360],{"className":22359},[55],[19,22361,22363,22408],{"className":22362},[126,127],[19,22364,22366,22405],{"className":22365},[131],[19,22367,22369,22383,22391],{"className":22368,"style":15704},[135],[19,22370,22371,22374],{"style":196},[19,22372],{"className":22373,"style":144},[143],[19,22375,22377],{"className":22376},[148,179,203,151],[19,22378,22380],{"className":22379},[155,151],[19,22381,63],{"className":22382},[155,151],[19,22384,22385,22388],{"style":212},[19,22386],{"className":22387,"style":144},[143],[19,22389],{"className":22390,"style":220},[219,151],[19,22392,22393,22396],{"style":223},[19,22394],{"className":22395,"style":144},[143],[19,22397,22399],{"className":22398},[148,179,203,151],[19,22400,22402],{"className":22401},[155,151],[19,22403,402],{"className":22404},[155,151],[19,22406,242],{"className":22407},[241],[19,22409,22411],{"className":22410},[131],[19,22412,22414],{"className":22413,"style":249},[135],[19,22415],{},[19,22417],{"className":22418},[255,178,148,179,180],[19,22420,22421,22424],{"style":3968},[19,22422],{"className":22423,"style":144},[143],[19,22425],{"className":22426,"style":3975},[219],[19,22428,22429,22432],{"style":8787},[19,22430],{"className":22431,"style":144},[143],[19,22433,22435],{"className":22434},[148,149,150,151],[19,22436,22438],{"className":22437},[155,151],[19,22439,402],{"className":22440},[155,151],[19,22442,242],{"className":22443},[241],[19,22445,22447],{"className":22446},[131],[19,22448,22451],{"className":22449,"style":22450},[135],"height:0.6418em;",[19,22452],{},[19,22454],{"className":22455},[255,178],[15,22457,22458],{},"Since:",[19,22460,22462],{"className":22461,"translate":23},[22],[19,22463,22465,22519],{"className":22464,"translate":23},[27],[19,22466,22468],{"className":22467},[31],[33,22469,22470],{"xmlns":35,"display":36},[38,22471,22472,22516],{},[41,22473,22474,22486,22488,22490,22492,22508,22510],{},[55,22475,22476,22478],{},[51,22477,402],{},[41,22479,22480,22482,22484],{},[58,22481,77],{},[47,22483,1066],{},[51,22485,402],{},[47,22487,8448],{},[58,22489,15927],{},[47,22491,74],{},[41,22493,22494,22496,22498,22500,22506],{},[47,22495,412],{"fence":99},[51,22497,402],{},[47,22499,1066],{},[55,22501,22502,22504],{},[51,22503,402],{},[58,22505,77],{},[47,22507,422],{"fence":99},[47,22509,8448],{},[55,22511,22512,22514],{},[51,22513,402],{},[58,22515,77],{},[91,22517,22518],{"encoding":93},"\\frac{1}{n + 1} \u003C \\ln \\left(1 + \\frac{1}{n}\\right) \u003C \\frac{1}{n}",[19,22520,22522,22611,22721],{"className":22521,"ariaHidden":99},[98],[19,22523,22525,22528,22602,22605,22608],{"className":22524},[103],[19,22526],{"className":22527,"style":11341},[107],[19,22529,22531,22534,22599],{"className":22530},[155],[19,22532],{"className":22533},[177,178],[19,22535,22537],{"className":22536},[55],[19,22538,22540,22591],{"className":22539},[126,127],[19,22541,22543,22588],{"className":22542},[131],[19,22544,22546,22569,22577],{"className":22545,"style":3953},[135],[19,22547,22548,22551],{"style":3956},[19,22549],{"className":22550,"style":144},[143],[19,22552,22554,22557,22560,22563,22566],{"className":22553},[155],[19,22555,77],{"className":22556},[155,236],[19,22558],{"className":22559,"style":920},[271],[19,22561,1066],{"className":22562},[788],[19,22564],{"className":22565,"style":920},[271],[19,22567,402],{"className":22568},[155],[19,22570,22571,22574],{"style":3968},[19,22572],{"className":22573,"style":144},[143],[19,22575],{"className":22576,"style":3975},[219],[19,22578,22579,22582],{"style":3978},[19,22580],{"className":22581,"style":144},[143],[19,22583,22585],{"className":22584},[155],[19,22586,402],{"className":22587},[155],[19,22589,242],{"className":22590},[241],[19,22592,22594],{"className":22593},[131],[19,22595,22597],{"className":22596,"style":5222},[135],[19,22598],{},[19,22600],{"className":22601},[255,178],[19,22603],{"className":22604,"style":607},[271],[19,22606,8448],{"className":22607},[611],[19,22609],{"className":22610,"style":607},[271],[19,22612,22614,22617,22620,22623,22712,22715,22718],{"className":22613},[103],[19,22615],{"className":22616,"style":10641},[107],[19,22618,15927],{"className":22619},[112],[19,22621],{"className":22622,"style":272},[271],[19,22624,22626,22632,22635,22638,22641,22644,22706],{"className":22625},[1353],[19,22627,22629],{"className":22628,"style":1358},[177,1357],[19,22630,412],{"className":22631},[1362,150],[19,22633,402],{"className":22634},[155],[19,22636],{"className":22637,"style":920},[271],[19,22639,1066],{"className":22640},[788],[19,22642],{"className":22643,"style":920},[271],[19,22645,22647,22650,22703],{"className":22646},[155],[19,22648],{"className":22649},[177,178],[19,22651,22653],{"className":22652},[55],[19,22654,22656,22695],{"className":22655},[126,127],[19,22657,22659,22692],{"className":22658},[131],[19,22660,22662,22673,22681],{"className":22661,"style":3953},[135],[19,22663,22664,22667],{"style":3956},[19,22665],{"className":22666,"style":144},[143],[19,22668,22670],{"className":22669},[155],[19,22671,77],{"className":22672},[155,236],[19,22674,22675,22678],{"style":3968},[19,22676],{"className":22677,"style":144},[143],[19,22679],{"className":22680,"style":3975},[219],[19,22682,22683,22686],{"style":3978},[19,22684],{"className":22685,"style":144},[143],[19,22687,22689],{"className":22688},[155],[19,22690,402],{"className":22691},[155],[19,22693,242],{"className":22694},[241],[19,22696,22698],{"className":22697},[131],[19,22699,22701],{"className":22700,"style":4009},[135],[19,22702],{},[19,22704],{"className":22705},[255,178],[19,22707,22709],{"className":22708,"style":1358},[255,1357],[19,22710,422],{"className":22711},[1362,150],[19,22713],{"className":22714,"style":607},[271],[19,22716,8448],{"className":22717},[611],[19,22719],{"className":22720,"style":607},[271],[19,22722,22724,22727],{"className":22723},[103],[19,22725],{"className":22726,"style":15770},[107],[19,22728,22730,22733,22786],{"className":22729},[155],[19,22731],{"className":22732},[177,178],[19,22734,22736],{"className":22735},[55],[19,22737,22739,22778],{"className":22738},[126,127],[19,22740,22742,22775],{"className":22741},[131],[19,22743,22745,22756,22764],{"className":22744,"style":3953},[135],[19,22746,22747,22750],{"style":3956},[19,22748],{"className":22749,"style":144},[143],[19,22751,22753],{"className":22752},[155],[19,22754,77],{"className":22755},[155,236],[19,22757,22758,22761],{"style":3968},[19,22759],{"className":22760,"style":144},[143],[19,22762],{"className":22763,"style":3975},[219],[19,22765,22766,22769],{"style":3978},[19,22767],{"className":22768,"style":144},[143],[19,22770,22772],{"className":22771},[155],[19,22773,402],{"className":22774},[155],[19,22776,242],{"className":22777},[241],[19,22779,22781],{"className":22780},[131],[19,22782,22784],{"className":22783,"style":4009},[135],[19,22785],{},[19,22787],{"className":22788},[255,178],[15,22790,22791,22794,22795,22885,22886,22979],{},[4202,22792,22793],{},"Note:"," The right side follows from the inequality ",[19,22796,22798,22828],{"className":22797,"translate":23},[27],[19,22799,22801],{"className":22800},[31],[33,22802,22803],{"xmlns":35},[38,22804,22805,22825],{},[41,22806,22807,22809,22811,22813,22815,22817,22819,22821,22823],{},[58,22808,15927],{},[47,22810,74],{},[47,22812,412],{"stretchy":411},[51,22814,402],{},[47,22816,1066],{},[58,22818,80],{},[47,22820,422],{"stretchy":411},[47,22822,8448],{},[58,22824,80],{},[91,22826,22827],{"encoding":93},"\\ln(1 + x) \u003C x",[19,22829,22831,22855,22876],{"className":22830,"ariaHidden":99},[98],[19,22832,22834,22837,22840,22843,22846,22849,22852],{"className":22833},[103],[19,22835],{"className":22836,"style":2167},[107],[19,22838,15927],{"className":22839},[112],[19,22841,412],{"className":22842},[177],[19,22844,402],{"className":22845},[155],[19,22847],{"className":22848,"style":920},[271],[19,22850,1066],{"className":22851},[788],[19,22853],{"className":22854,"style":920},[271],[19,22856,22858,22861,22864,22867,22870,22873],{"className":22857},[103],[19,22859],{"className":22860,"style":2167},[107],[19,22862,80],{"className":22863},[155,236],[19,22865,422],{"className":22866},[255],[19,22868],{"className":22869,"style":607},[271],[19,22871,8448],{"className":22872},[611],[19,22874],{"className":22875,"style":607},[271],[19,22877,22879,22882],{"className":22878},[103],[19,22880],{"className":22881,"style":4230},[107],[19,22883,80],{"className":22884},[155,236]," (for ",[19,22887,22889,22908],{"className":22888,"translate":23},[27],[19,22890,22892],{"className":22891},[31],[33,22893,22894],{"xmlns":35},[38,22895,22896,22905],{},[41,22897,22898,22900,22903],{},[58,22899,80],{},[47,22901,22902],{"mathvariant":10345},"≠",[51,22904,53],{},[91,22906,22907],{"encoding":93},"x \\neq 0",[19,22909,22911,22970],{"className":22910,"ariaHidden":99},[98],[19,22912,22914,22918,22921,22924,22967],{"className":22913},[103],[19,22915],{"className":22916,"style":22917},[107],"height:0.8889em;vertical-align:-0.1944em;",[19,22919,80],{"className":22920},[155,236],[19,22922],{"className":22923,"style":607},[271],[19,22925,22927,22960,22964],{"className":22926},[611],[19,22928,22930],{"className":22929},[611],[19,22931,22934],{"className":22932},[155,22933],"vbox",[19,22935,22938],{"className":22936},[22937],"thinbox",[19,22939,22942,22945,22956],{"className":22940},[22941],"rlap",[19,22943],{"className":22944,"style":22917},[107],[19,22946,22949],{"className":22947},[22948],"inner",[19,22950,22952],{"className":22951},[155],[19,22953,22955],{"className":22954},[611],"",[19,22957],{"className":22958},[22959],"fix",[19,22961],{"className":22962},[271,22963],"nobreak",[19,22965,370],{"className":22966},[611],[19,22968],{"className":22969,"style":607},[271],[19,22971,22973,22976],{"className":22972},[103],[19,22974],{"className":22975,"style":5939},[107],[19,22977,53],{"className":22978},[155],").",[15,22981,22982,22985,22986,23254,23255,23365],{},[4202,22983,22984],{},"The left side follows from:"," ",[19,22987,22989,23041],{"className":22988,"translate":23},[27],[19,22990,22992],{"className":22991},[31],[33,22993,22994],{"xmlns":35},[38,22995,22996,23038],{},[41,22997,22998,23000,23002,23014,23016,23018,23020],{},[51,22999,402],{},[47,23001,373],{},[55,23003,23004,23006],{},[58,23005,77],{},[41,23007,23008,23010,23012],{},[58,23009,77],{},[47,23011,1066],{},[51,23013,402],{},[47,23015,8448],{},[58,23017,15927],{},[47,23019,74],{},[41,23021,23022,23024,23036],{},[47,23023,412],{"fence":99},[55,23025,23026,23034],{},[41,23027,23028,23030,23032],{},[58,23029,77],{},[47,23031,1066],{},[51,23033,402],{},[58,23035,77],{},[47,23037,422],{"fence":99},[91,23039,23040],{"encoding":93},"1 - \\frac{n}{n+1} \u003C \\ln \\left(\\frac{n+1}{n}\\right)",[19,23042,23044,23062,23152],{"className":23043,"ariaHidden":99},[98],[19,23045,23047,23050,23053,23056,23059],{"className":23046},[103],[19,23048],{"className":23049,"style":16847},[107],[19,23051,402],{"className":23052},[155],[19,23054],{"className":23055,"style":920},[271],[19,23057,373],{"className":23058},[788],[19,23060],{"className":23061,"style":920},[271],[19,23063,23065,23069,23143,23146,23149],{"className":23064},[103],[19,23066],{"className":23067,"style":23068},[107],"height:1.0987em;vertical-align:-0.4033em;",[19,23070,23072,23075,23140],{"className":23071},[155],[19,23073],{"className":23074},[177,178],[19,23076,23078],{"className":23077},[55],[19,23079,23081,23132],{"className":23080},[126,127],[19,23082,23084,23129],{"className":23083},[131],[19,23085,23087,23107,23115],{"className":23086,"style":8761},[135],[19,23088,23089,23092],{"style":8764},[19,23090],{"className":23091,"style":144},[143],[19,23093,23095],{"className":23094},[148,149,150,151],[19,23096,23098,23101,23104],{"className":23097},[155,151],[19,23099,77],{"className":23100},[155,236,151],[19,23102,1066],{"className":23103},[788,151],[19,23105,402],{"className":23106},[155,151],[19,23108,23109,23112],{"style":3968},[19,23110],{"className":23111,"style":144},[143],[19,23113],{"className":23114,"style":3975},[219],[19,23116,23117,23120],{"style":8787},[19,23118],{"className":23119,"style":144},[143],[19,23121,23123],{"className":23122},[148,149,150,151],[19,23124,23126],{"className":23125},[155,151],[19,23127,77],{"className":23128},[155,236,151],[19,23130,242],{"className":23131},[241],[19,23133,23135],{"className":23134},[131],[19,23136,23138],{"className":23137,"style":11748},[135],[19,23139],{},[19,23141],{"className":23142},[255,178],[19,23144],{"className":23145,"style":607},[271],[19,23147,8448],{"className":23148},[611],[19,23150],{"className":23151,"style":607},[271],[19,23153,23155,23159,23162,23165],{"className":23154},[103],[19,23156],{"className":23157,"style":23158},[107],"height:1.2em;vertical-align:-0.35em;",[19,23160,15927],{"className":23161},[112],[19,23163],{"className":23164,"style":272},[271],[19,23166,23168,23174,23248],{"className":23167},[1353],[19,23169,23171],{"className":23170,"style":1358},[177,1357],[19,23172,412],{"className":23173},[1362,203],[19,23175,23177,23180,23245],{"className":23176},[155],[19,23178],{"className":23179},[177,178],[19,23181,23183],{"className":23182},[55],[19,23184,23186,23237],{"className":23185},[126,127],[19,23187,23189,23234],{"className":23188},[131],[19,23190,23192,23206,23214],{"className":23191,"style":19415},[135],[19,23193,23194,23197],{"style":8764},[19,23195],{"className":23196,"style":144},[143],[19,23198,23200],{"className":23199},[148,149,150,151],[19,23201,23203],{"className":23202},[155,151],[19,23204,77],{"className":23205},[155,236,151],[19,23207,23208,23211],{"style":3968},[19,23209],{"className":23210,"style":144},[143],[19,23212],{"className":23213,"style":3975},[219],[19,23215,23216,23219],{"style":8787},[19,23217],{"className":23218,"style":144},[143],[19,23220,23222],{"className":23221},[148,149,150,151],[19,23223,23225,23228,23231],{"className":23224},[155,151],[19,23226,77],{"className":23227},[155,236,151],[19,23229,1066],{"className":23230},[788,151],[19,23232,402],{"className":23233},[155,151],[19,23235,242],{"className":23236},[241],[19,23238,23240],{"className":23239},[131],[19,23241,23243],{"className":23242,"style":8809},[135],[19,23244],{},[19,23246],{"className":23247},[255,178],[19,23249,23251],{"className":23250,"style":1358},[255,1357],[19,23252,422],{"className":23253},[1362,203],", where ",[19,23256,23258,23282],{"className":23257,"translate":23},[27],[19,23259,23261],{"className":23260},[31],[33,23262,23263],{"xmlns":35},[38,23264,23265,23279],{},[41,23266,23267],{},[55,23268,23269,23271],{},[58,23270,77],{},[41,23272,23273,23275,23277],{},[58,23274,77],{},[47,23276,1066],{},[51,23278,402],{},[91,23280,23281],{"encoding":93},"\\frac{n}{n+1}",[19,23283,23285],{"className":23284,"ariaHidden":99},[98],[19,23286,23288,23291],{"className":23287},[103],[19,23289],{"className":23290,"style":23068},[107],[19,23292,23294,23297,23362],{"className":23293},[155],[19,23295],{"className":23296},[177,178],[19,23298,23300],{"className":23299},[55],[19,23301,23303,23354],{"className":23302},[126,127],[19,23304,23306,23351],{"className":23305},[131],[19,23307,23309,23329,23337],{"className":23308,"style":8761},[135],[19,23310,23311,23314],{"style":8764},[19,23312],{"className":23313,"style":144},[143],[19,23315,23317],{"className":23316},[148,149,150,151],[19,23318,23320,23323,23326],{"className":23319},[155,151],[19,23321,77],{"className":23322},[155,236,151],[19,23324,1066],{"className":23325},[788,151],[19,23327,402],{"className":23328},[155,151],[19,23330,23331,23334],{"style":3968},[19,23332],{"className":23333,"style":144},[143],[19,23335],{"className":23336,"style":3975},[219],[19,23338,23339,23342],{"style":8787},[19,23340],{"className":23341,"style":144},[143],[19,23343,23345],{"className":23344},[148,149,150,151],[19,23346,23348],{"className":23347},[155,151],[19,23349,77],{"className":23350},[155,236,151],[19,23352,242],{"className":23353},[241],[19,23355,23357],{"className":23356},[131],[19,23358,23360],{"className":23359,"style":11748},[135],[19,23361],{},[19,23363],{"className":23364},[255,178]," is considered as a whole. This is high school knowledge.",[15,23367,23368,23369,23467],{},"Because ",[19,23370,23372,23390],{"className":23371,"translate":23},[27],[19,23373,23375],{"className":23374},[31],[33,23376,23377],{"xmlns":35},[38,23378,23379,23387],{},[41,23380,23381],{},[55,23382,23383,23385],{},[51,23384,402],{},[58,23386,80],{},[91,23388,23389],{"encoding":93},"\\frac{1}{x}",[19,23391,23393],{"className":23392,"ariaHidden":99},[98],[19,23394,23396,23399],{"className":23395},[103],[19,23397],{"className":23398,"style":22192},[107],[19,23400,23402,23405,23464],{"className":23401},[155],[19,23403],{"className":23404},[177,178],[19,23406,23408],{"className":23407},[55],[19,23409,23411,23456],{"className":23410},[126,127],[19,23412,23414,23453],{"className":23413},[131],[19,23415,23417,23431,23439],{"className":23416,"style":19415},[135],[19,23418,23419,23422],{"style":8764},[19,23420],{"className":23421,"style":144},[143],[19,23423,23425],{"className":23424},[148,149,150,151],[19,23426,23428],{"className":23427},[155,151],[19,23429,80],{"className":23430},[155,236,151],[19,23432,23433,23436],{"style":3968},[19,23434],{"className":23435,"style":144},[143],[19,23437],{"className":23438,"style":3975},[219],[19,23440,23441,23444],{"style":8787},[19,23442],{"className":23443,"style":144},[143],[19,23445,23447],{"className":23446},[148,149,150,151],[19,23448,23450],{"className":23449},[155,151],[19,23451,402],{"className":23452},[155,151],[19,23454,242],{"className":23455},[241],[19,23457,23459],{"className":23458},[131],[19,23460,23462],{"className":23461,"style":8809},[135],[19,23463],{},[19,23465],{"className":23466},[255,178]," is a concave function:",[19,23469,23471],{"className":23470,"translate":23},[22],[19,23472,23474,23548],{"className":23473,"translate":23},[27],[19,23475,23477],{"className":23476},[31],[33,23478,23479],{"xmlns":35,"display":36},[38,23480,23481,23545],{},[41,23482,23483,23497,23503,23505,23507,23509,23511,23513,23515,23527,23529],{},[44,23484,23485,23487,23489],{},[47,23486,49],{},[58,23488,77],{},[41,23490,23491,23493,23495],{},[58,23492,77],{},[47,23494,1066],{},[51,23496,402],{},[55,23498,23499,23501],{},[51,23500,402],{},[58,23502,80],{},[82,23504,84],{},[58,23506,87],{},[58,23508,80],{},[47,23510,370],{},[58,23512,15927],{},[47,23514,74],{},[55,23516,23517,23525],{},[41,23518,23519,23521,23523],{},[58,23520,77],{},[47,23522,1066],{},[51,23524,402],{},[58,23526,77],{},[47,23528,19112],{},[55,23530,23531,23533],{},[51,23532,402],{},[41,23534,23535,23537,23539],{},[58,23536,77],{},[47,23538,1066],{},[55,23540,23541,23543],{},[51,23542,402],{},[51,23544,63],{},[91,23546,23547],{"encoding":93},"\\int_{n}^{n+1} \\frac{1}{x} \\, dx = \\ln \\frac{n+1}{n} > \\frac{1}{n + \\frac{1}{2}}",[19,23549,23551,23703,23798],{"className":23550,"ariaHidden":99},[98],[19,23552,23554,23557,23620,23623,23685,23688,23691,23694,23697,23700],{"className":23553},[103],[19,23555],{"className":23556,"style":17701},[107],[19,23558,23560,23563],{"className":23559},[112],[19,23561,49],{"className":23562,"style":118},[112,116,117],[19,23564,23566],{"className":23565},[122],[19,23567,23569,23612],{"className":23568},[126,127],[19,23570,23572,23609],{"className":23571},[131],[19,23573,23575,23589],{"className":23574,"style":17720},[135],[19,23576,23577,23580],{"style":17723},[19,23578],{"className":23579,"style":298},[143],[19,23581,23583],{"className":23582},[148,149,150,151],[19,23584,23586],{"className":23585},[155,151],[19,23587,77],{"className":23588},[155,236,151],[19,23590,23591,23594],{"style":17738},[19,23592],{"className":23593,"style":298},[143],[19,23595,23597],{"className":23596},[148,149,150,151],[19,23598,23600,23603,23606],{"className":23599},[155,151],[19,23601,77],{"className":23602},[155,236,151],[19,23604,1066],{"className":23605},[788,151],[19,23607,402],{"className":23608},[155,151],[19,23610,242],{"className":23611},[241],[19,23613,23615],{"className":23614},[131],[19,23616,23618],{"className":23617,"style":265},[135],[19,23619],{},[19,23621],{"className":23622,"style":272},[271],[19,23624,23626,23629,23682],{"className":23625},[155],[19,23627],{"className":23628},[177,178],[19,23630,23632],{"className":23631},[55],[19,23633,23635,23674],{"className":23634},[126,127],[19,23636,23638,23671],{"className":23637},[131],[19,23639,23641,23652,23660],{"className":23640,"style":3953},[135],[19,23642,23643,23646],{"style":3956},[19,23644],{"className":23645,"style":144},[143],[19,23647,23649],{"className":23648},[155],[19,23650,80],{"className":23651},[155,236],[19,23653,23654,23657],{"style":3968},[19,23655],{"className":23656,"style":144},[143],[19,23658],{"className":23659,"style":3975},[219],[19,23661,23662,23665],{"style":3978},[19,23663],{"className":23664,"style":144},[143],[19,23666,23668],{"className":23667},[155],[19,23669,402],{"className":23670},[155],[19,23672,242],{"className":23673},[241],[19,23675,23677],{"className":23676},[131],[19,23678,23680],{"className":23679,"style":4009},[135],[19,23681],{},[19,23683],{"className":23684},[255,178],[19,23686],{"className":23687,"style":272},[271],[19,23689,87],{"className":23690},[155,236],[19,23692,80],{"className":23693},[155,236],[19,23695],{"className":23696,"style":607},[271],[19,23698,370],{"className":23699},[611],[19,23701],{"className":23702,"style":607},[271],[19,23704,23706,23709,23712,23715,23789,23792,23795],{"className":23705},[103],[19,23707],{"className":23708,"style":15770},[107],[19,23710,15927],{"className":23711},[112],[19,23713],{"className":23714,"style":272},[271],[19,23716,23718,23721,23786],{"className":23717},[155],[19,23719],{"className":23720},[177,178],[19,23722,23724],{"className":23723},[55],[19,23725,23727,23778],{"className":23726},[126,127],[19,23728,23730,23775],{"className":23729},[131],[19,23731,23733,23744,23752],{"className":23732,"style":3953},[135],[19,23734,23735,23738],{"style":3956},[19,23736],{"className":23737,"style":144},[143],[19,23739,23741],{"className":23740},[155],[19,23742,77],{"className":23743},[155,236],[19,23745,23746,23749],{"style":3968},[19,23747],{"className":23748,"style":144},[143],[19,23750],{"className":23751,"style":3975},[219],[19,23753,23754,23757],{"style":3978},[19,23755],{"className":23756,"style":144},[143],[19,23758,23760,23763,23766,23769,23772],{"className":23759},[155],[19,23761,77],{"className":23762},[155,236],[19,23764],{"className":23765,"style":920},[271],[19,23767,1066],{"className":23768},[788],[19,23770],{"className":23771,"style":920},[271],[19,23773,402],{"className":23774},[155],[19,23776,242],{"className":23777},[241],[19,23779,23781],{"className":23780},[131],[19,23782,23784],{"className":23783,"style":4009},[135],[19,23785],{},[19,23787],{"className":23788},[255,178],[19,23790],{"className":23791,"style":607},[271],[19,23793,19112],{"className":23794},[611],[19,23796],{"className":23797,"style":607},[271],[19,23799,23801,23805],{"className":23800},[103],[19,23802],{"className":23803,"style":23804},[107],"height:2.4015em;vertical-align:-1.0801em;",[19,23806,23808,23811,23943],{"className":23807},[155],[19,23809],{"className":23810},[177,178],[19,23812,23814],{"className":23813},[55],[19,23815,23817,23934],{"className":23816},[126,127],[19,23818,23820,23931],{"className":23819},[131],[19,23821,23823,23912,23920],{"className":23822,"style":3953},[135],[19,23824,23826,23829],{"style":23825},"top:-2.2649em;",[19,23827],{"className":23828,"style":144},[143],[19,23830,23832,23835,23838,23841,23844],{"className":23831},[155],[19,23833,77],{"className":23834},[155,236],[19,23836],{"className":23837,"style":920},[271],[19,23839,1066],{"className":23840},[788],[19,23842],{"className":23843,"style":920},[271],[19,23845,23847,23850,23909],{"className":23846},[155],[19,23848],{"className":23849},[177,178],[19,23851,23853],{"className":23852},[55],[19,23854,23856,23901],{"className":23855},[126,127],[19,23857,23859,23898],{"className":23858},[131],[19,23860,23862,23876,23884],{"className":23861,"style":19415},[135],[19,23863,23864,23867],{"style":8764},[19,23865],{"className":23866,"style":144},[143],[19,23868,23870],{"className":23869},[148,149,150,151],[19,23871,23873],{"className":23872},[155,151],[19,23874,63],{"className":23875},[155,151],[19,23877,23878,23881],{"style":3968},[19,23879],{"className":23880,"style":144},[143],[19,23882],{"className":23883,"style":3975},[219],[19,23885,23886,23889],{"style":8787},[19,23887],{"className":23888,"style":144},[143],[19,23890,23892],{"className":23891},[148,149,150,151],[19,23893,23895],{"className":23894},[155,151],[19,23896,402],{"className":23897},[155,151],[19,23899,242],{"className":23900},[241],[19,23902,23904],{"className":23903},[131],[19,23905,23907],{"className":23906,"style":8809},[135],[19,23908],{},[19,23910],{"className":23911},[255,178],[19,23913,23914,23917],{"style":3968},[19,23915],{"className":23916,"style":144},[143],[19,23918],{"className":23919,"style":3975},[219],[19,23921,23922,23925],{"style":3978},[19,23923],{"className":23924,"style":144},[143],[19,23926,23928],{"className":23927},[155],[19,23929,402],{"className":23930},[155],[19,23932,242],{"className":23933},[241],[19,23935,23937],{"className":23936},[131],[19,23938,23941],{"className":23939,"style":23940},[135],"height:1.0801em;",[19,23942],{},[19,23944],{"className":23945},[255,178],[15,23947,4198],{},[19,23949,23951],{"className":23950,"translate":23},[22],[19,23952,23954,24042],{"className":23953,"translate":23},[27],[19,23955,23957],{"className":23956},[31],[33,23958,23959],{"xmlns":35,"display":36},[38,23960,23961,24039],{},[41,23962,23963,23979,23999,24001,24035,24037],{},[10327,23964,23965,23971],{},[41,23966,23967,23969],{},[58,23968,10333],{},[47,23970,74],{},[41,23972,23973,23975,23977],{},[58,23974,77],{},[47,23976,10342],{},[58,23978,10346],{"mathvariant":10345},[55,23980,23981,23993],{},[18497,23982,23983,23985],{},[58,23984,18501],{},[41,23986,23987,23989,23991],{},[58,23988,77],{},[47,23990,1066],{},[51,23992,402],{},[18497,23994,23995,23997],{},[58,23996,18501],{},[58,23998,77],{},[47,24000,370],{},[55,24002,24003,24005],{},[58,24004,15064],{},[65,24006,24007,24023],{},[41,24008,24009,24011,24013,24015,24021],{},[47,24010,412],{"fence":99},[51,24012,402],{},[47,24014,1066],{},[55,24016,24017,24019],{},[51,24018,402],{},[58,24020,77],{},[47,24022,422],{"fence":99},[41,24024,24025,24027,24029],{},[58,24026,77],{},[47,24028,1066],{},[55,24030,24031,24033],{},[51,24032,402],{},[51,24034,63],{},[47,24036,8448],{},[51,24038,402],{},[91,24040,24041],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}} \u003C 1",[19,24043,24045,24261,24530],{"className":24044,"ariaHidden":99},[98],[19,24046,24048,24051,24104,24107,24252,24255,24258],{"className":24047},[103],[19,24049],{"className":24050,"style":19127},[107],[19,24052,24054],{"className":24053},[112,10645],[19,24055,24057,24096],{"className":24056},[126,127],[19,24058,24060,24093],{"className":24059},[131],[19,24061,24063,24083],{"className":24062,"style":4286},[135],[19,24064,24065,24068],{"style":15507},[19,24066],{"className":24067,"style":144},[143],[19,24069,24071],{"className":24070},[148,149,150,151],[19,24072,24074,24077,24080],{"className":24073},[155,151],[19,24075,77],{"className":24076},[155,236,151],[19,24078,10342],{"className":24079},[611,151],[19,24081,10346],{"className":24082},[155,151],[19,24084,24085,24088],{"style":10678},[19,24086],{"className":24087,"style":144},[143],[19,24089,24090],{},[19,24091,10333],{"className":24092},[112],[19,24094,242],{"className":24095},[241],[19,24097,24099],{"className":24098},[131],[19,24100,24102],{"className":24101,"style":15545},[135],[19,24103],{},[19,24105],{"className":24106,"style":272},[271],[19,24108,24110,24113,24249],{"className":24109},[155],[19,24111],{"className":24112},[177,178],[19,24114,24116],{"className":24115},[55],[19,24117,24119,24241],{"className":24118},[126,127],[19,24120,24122,24238],{"className":24121},[131],[19,24123,24125,24173,24181],{"className":24124,"style":5785},[135],[19,24126,24127,24130],{"style":3956},[19,24128],{"className":24129,"style":144},[143],[19,24131,24133],{"className":24132},[155],[19,24134,24136,24139],{"className":24135},[155],[19,24137,18501],{"className":24138},[155,236],[19,24140,24142],{"className":24141},[122],[19,24143,24145,24165],{"className":24144},[126,127],[19,24146,24148,24162],{"className":24147},[131],[19,24149,24151],{"className":24150,"style":18626},[135],[19,24152,24153,24156],{"style":18629},[19,24154],{"className":24155,"style":298},[143],[19,24157,24159],{"className":24158},[148,149,150,151],[19,24160,77],{"className":24161},[155,236,151],[19,24163,242],{"className":24164},[241],[19,24166,24168],{"className":24167},[131],[19,24169,24171],{"className":24170,"style":18648},[135],[19,24172],{},[19,24174,24175,24178],{"style":3968},[19,24176],{"className":24177,"style":144},[143],[19,24179],{"className":24180,"style":3975},[219],[19,24182,24183,24186],{"style":3978},[19,24184],{"className":24185,"style":144},[143],[19,24187,24189],{"className":24188},[155],[19,24190,24192,24195],{"className":24191},[155],[19,24193,18501],{"className":24194},[155,236],[19,24196,24198],{"className":24197},[122],[19,24199,24201,24230],{"className":24200},[126,127],[19,24202,24204,24227],{"className":24203},[131],[19,24205,24207],{"className":24206,"style":18814},[135],[19,24208,24209,24212],{"style":18629},[19,24210],{"className":24211,"style":298},[143],[19,24213,24215],{"className":24214},[148,149,150,151],[19,24216,24218,24221,24224],{"className":24217},[155,151],[19,24219,77],{"className":24220},[155,236,151],[19,24222,1066],{"className":24223},[788,151],[19,24225,402],{"className":24226},[155,151],[19,24228,242],{"className":24229},[241],[19,24231,24233],{"className":24232},[131],[19,24234,24236],{"className":24235,"style":18844},[135],[19,24237],{},[19,24239,242],{"className":24240},[241],[19,24242,24244],{"className":24243},[131],[19,24245,24247],{"className":24246,"style":19324},[135],[19,24248],{},[19,24250],{"className":24251},[255,178],[19,24253],{"className":24254,"style":607},[271],[19,24256,370],{"className":24257},[611],[19,24259],{"className":24260,"style":607},[271],[19,24262,24264,24267,24521,24524,24527],{"className":24263},[103],[19,24265],{"className":24266,"style":21090},[107],[19,24268,24270,24273,24518],{"className":24269},[155],[19,24271],{"className":24272},[177,178],[19,24274,24276],{"className":24275},[55],[19,24277,24279,24510],{"className":24278},[126,127],[19,24280,24282,24507],{"className":24281},[131],[19,24283,24285,24488,24496],{"className":24284,"style":5785},[135],[19,24286,24287,24290],{"style":21111},[19,24288],{"className":24289,"style":21115},[143],[19,24291,24293],{"className":24292},[155],[19,24294,24296,24391],{"className":24295},[1353],[19,24297,24299,24305,24308,24311,24314,24317,24385],{"className":24298},[1353],[19,24300,24302],{"className":24301,"style":1358},[177,1357],[19,24303,412],{"className":24304},[1362,203],[19,24306,402],{"className":24307},[155],[19,24309],{"className":24310,"style":920},[271],[19,24312,1066],{"className":24313},[788],[19,24315],{"className":24316,"style":920},[271],[19,24318,24320,24323,24382],{"className":24319},[155],[19,24321],{"className":24322},[177,178],[19,24324,24326],{"className":24325},[55],[19,24327,24329,24374],{"className":24328},[126,127],[19,24330,24332,24371],{"className":24331},[131],[19,24333,24335,24349,24357],{"className":24334,"style":19415},[135],[19,24336,24337,24340],{"style":8764},[19,24338],{"className":24339,"style":144},[143],[19,24341,24343],{"className":24342},[148,149,150,151],[19,24344,24346],{"className":24345},[155,151],[19,24347,77],{"className":24348},[155,236,151],[19,24350,24351,24354],{"style":3968},[19,24352],{"className":24353,"style":144},[143],[19,24355],{"className":24356,"style":3975},[219],[19,24358,24359,24362],{"style":8787},[19,24360],{"className":24361,"style":144},[143],[19,24363,24365],{"className":24364},[148,149,150,151],[19,24366,24368],{"className":24367},[155,151],[19,24369,402],{"className":24370},[155,151],[19,24372,242],{"className":24373},[241],[19,24375,24377],{"className":24376},[131],[19,24378,24380],{"className":24379,"style":8809},[135],[19,24381],{},[19,24383],{"className":24384},[255,178],[19,24386,24388],{"className":24387,"style":1358},[255,1357],[19,24389,422],{"className":24390},[1362,203],[19,24392,24394],{"className":24393},[122],[19,24395,24397],{"className":24396},[126],[19,24398,24400],{"className":24399},[131],[19,24401,24403],{"className":24402,"style":21229},[135],[19,24404,24405,24408],{"style":17908},[19,24406],{"className":24407,"style":144},[143],[19,24409,24411],{"className":24410},[148,149,150,151],[19,24412,24414,24417,24420],{"className":24413},[155,151],[19,24415,77],{"className":24416},[155,236,151],[19,24418,1066],{"className":24419},[788,151],[19,24421,24423,24426,24485],{"className":24422},[155,151],[19,24424],{"className":24425},[177,178,148,179,180],[19,24427,24429],{"className":24428},[55],[19,24430,24432,24477],{"className":24431},[126,127],[19,24433,24435,24474],{"className":24434},[131],[19,24436,24438,24452,24460],{"className":24437,"style":15704},[135],[19,24439,24440,24443],{"style":196},[19,24441],{"className":24442,"style":144},[143],[19,24444,24446],{"className":24445},[148,179,203,151],[19,24447,24449],{"className":24448},[155,151],[19,24450,63],{"className":24451},[155,151],[19,24453,24454,24457],{"style":212},[19,24455],{"className":24456,"style":144},[143],[19,24458],{"className":24459,"style":220},[219,151],[19,24461,24462,24465],{"style":223},[19,24463],{"className":24464,"style":144},[143],[19,24466,24468],{"className":24467},[148,179,203,151],[19,24469,24471],{"className":24470},[155,151],[19,24472,402],{"className":24473},[155,151],[19,24475,242],{"className":24476},[241],[19,24478,24480],{"className":24479},[131],[19,24481,24483],{"className":24482,"style":249},[135],[19,24484],{},[19,24486],{"className":24487},[255,178,148,179,180],[19,24489,24490,24493],{"style":21317},[19,24491],{"className":24492,"style":21115},[143],[19,24494],{"className":24495,"style":3975},[219],[19,24497,24498,24501],{"style":21326},[19,24499],{"className":24500,"style":21115},[143],[19,24502,24504],{"className":24503},[155],[19,24505,15064],{"className":24506},[155,236],[19,24508,242],{"className":24509},[241],[19,24511,24513],{"className":24512},[131],[19,24514,24516],{"className":24515,"style":21345},[135],[19,24517],{},[19,24519],{"className":24520},[255,178],[19,24522],{"className":24523,"style":607},[271],[19,24525,8448],{"className":24526},[611],[19,24528],{"className":24529,"style":607},[271],[19,24531,24533,24536],{"className":24532},[103],[19,24534],{"className":24535,"style":5939},[107],[19,24537,402],{"className":24538},[155],[15,24540,24541,24542,24611,24612,24705],{},"Thus, ",[19,24543,24545,24562],{"className":24544,"translate":23},[27],[19,24546,24548],{"className":24547},[31],[33,24549,24550],{"xmlns":35},[38,24551,24552,24560],{},[41,24553,24554],{},[18497,24555,24556,24558],{},[58,24557,18501],{},[58,24559,77],{},[91,24561,19628],{"encoding":93},[19,24563,24565],{"className":24564,"ariaHidden":99},[98],[19,24566,24568,24571],{"className":24567},[103],[19,24569],{"className":24570,"style":18607},[107],[19,24572,24574,24577],{"className":24573},[155],[19,24575,18501],{"className":24576},[155,236],[19,24578,24580],{"className":24579},[122],[19,24581,24583,24603],{"className":24582},[126,127],[19,24584,24586,24600],{"className":24585},[131],[19,24587,24589],{"className":24588,"style":18626},[135],[19,24590,24591,24594],{"style":18629},[19,24592],{"className":24593,"style":298},[143],[19,24595,24597],{"className":24596},[148,149,150,151],[19,24598,77],{"className":24599},[155,236,151],[19,24601,242],{"className":24602},[241],[19,24604,24606],{"className":24605},[131],[19,24607,24609],{"className":24608,"style":18648},[135],[19,24610],{}," is monotonically decreasing and ",[19,24613,24615,24637],{"className":24614,"translate":23},[27],[19,24616,24618],{"className":24617},[31],[33,24619,24620],{"xmlns":35},[38,24621,24622,24634],{},[41,24623,24624,24630,24632],{},[18497,24625,24626,24628],{},[58,24627,18501],{},[58,24629,77],{},[47,24631,19112],{},[51,24633,53],{},[91,24635,24636],{"encoding":93},"a_n > 0",[19,24638,24640,24696],{"className":24639,"ariaHidden":99},[98],[19,24641,24643,24647,24687,24690,24693],{"className":24642},[103],[19,24644],{"className":24645,"style":24646},[107],"height:0.6891em;vertical-align:-0.15em;",[19,24648,24650,24653],{"className":24649},[155],[19,24651,18501],{"className":24652},[155,236],[19,24654,24656],{"className":24655},[122],[19,24657,24659,24679],{"className":24658},[126,127],[19,24660,24662,24676],{"className":24661},[131],[19,24663,24665],{"className":24664,"style":18626},[135],[19,24666,24667,24670],{"style":18629},[19,24668],{"className":24669,"style":298},[143],[19,24671,24673],{"className":24672},[148,149,150,151],[19,24674,77],{"className":24675},[155,236,151],[19,24677,242],{"className":24678},[241],[19,24680,24682],{"className":24681},[131],[19,24683,24685],{"className":24684,"style":18648},[135],[19,24686],{},[19,24688],{"className":24689,"style":607},[271],[19,24691,19112],{"className":24692},[611],[19,24694],{"className":24695,"style":607},[271],[19,24697,24699,24702],{"className":24698},[103],[19,24700],{"className":24701,"style":5939},[107],[19,24703,53],{"className":24704},[155],", so it must converge, i.e., it has a limit.",[15,24707,24708],{},"Let:",[19,24710,24712],{"className":24711,"translate":23},[22],[19,24713,24715,24777],{"className":24714,"translate":23},[27],[19,24716,24718],{"className":24717},[31],[33,24719,24720],{"xmlns":35,"display":36},[38,24721,24722,24774],{},[41,24723,24724,24740,24764,24770,24772],{},[10327,24725,24726,24732],{},[41,24727,24728,24730],{},[58,24729,10333],{},[47,24731,74],{},[41,24733,24734,24736,24738],{},[58,24735,77],{},[47,24737,10342],{},[58,24739,10346],{"mathvariant":10345},[55,24741,24742,24748],{},[41,24743,24744,24746],{},[58,24745,77],{},[47,24747,7425],{"stretchy":411},[65,24749,24750,24752],{},[58,24751,77],{},[41,24753,24754,24756,24758],{},[58,24755,77],{},[47,24757,1066],{},[55,24759,24760,24762],{},[51,24761,402],{},[51,24763,63],{},[65,24765,24766,24768],{},[58,24767,15064],{},[58,24769,77],{},[47,24771,370],{},[58,24773,18501],{},[91,24775,24776],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^{n + \\frac{1}{2}}} e^n = a",[19,24778,24780,25045],{"className":24779,"ariaHidden":99},[98],[19,24781,24783,24786,24839,24842,25007,25036,25039,25042],{"className":24782},[103],[19,24784],{"className":24785,"style":19802},[107],[19,24787,24789],{"className":24788},[112,10645],[19,24790,24792,24831],{"className":24791},[126,127],[19,24793,24795,24828],{"className":24794},[131],[19,24796,24798,24818],{"className":24797,"style":4286},[135],[19,24799,24800,24803],{"style":15507},[19,24801],{"className":24802,"style":144},[143],[19,24804,24806],{"className":24805},[148,149,150,151],[19,24807,24809,24812,24815],{"className":24808},[155,151],[19,24810,77],{"className":24811},[155,236,151],[19,24813,10342],{"className":24814},[611,151],[19,24816,10346],{"className":24817},[155,151],[19,24819,24820,24823],{"style":10678},[19,24821],{"className":24822,"style":144},[143],[19,24824,24825],{},[19,24826,10333],{"className":24827},[112],[19,24829,242],{"className":24830},[241],[19,24832,24834],{"className":24833},[131],[19,24835,24837],{"className":24836,"style":15545},[135],[19,24838],{},[19,24840],{"className":24841,"style":272},[271],[19,24843,24845,24848,25004],{"className":24844},[155],[19,24846],{"className":24847},[177,178],[19,24849,24851],{"className":24850},[55],[19,24852,24854,24996],{"className":24853},[126,127],[19,24855,24857,24993],{"className":24856},[131],[19,24858,24860,24971,24979],{"className":24859,"style":4771},[135],[19,24861,24862,24865],{"style":19823},[19,24863],{"className":24864,"style":144},[143],[19,24866,24868],{"className":24867},[155],[19,24869,24871,24874],{"className":24870},[155],[19,24872,77],{"className":24873},[155,236],[19,24875,24877],{"className":24876},[122],[19,24878,24880],{"className":24879},[126],[19,24881,24883],{"className":24882},[131],[19,24884,24886],{"className":24885,"style":19848},[135],[19,24887,24888,24891],{"style":19851},[19,24889],{"className":24890,"style":144},[143],[19,24892,24894],{"className":24893},[148,149,150,151],[19,24895,24897,24900,24903],{"className":24896},[155,151],[19,24898,77],{"className":24899},[155,236,151],[19,24901,1066],{"className":24902},[788,151],[19,24904,24906,24909,24968],{"className":24905},[155,151],[19,24907],{"className":24908},[177,178,148,179,180],[19,24910,24912],{"className":24911},[55],[19,24913,24915,24960],{"className":24914},[126,127],[19,24916,24918,24957],{"className":24917},[131],[19,24919,24921,24935,24943],{"className":24920,"style":15704},[135],[19,24922,24923,24926],{"style":196},[19,24924],{"className":24925,"style":144},[143],[19,24927,24929],{"className":24928},[148,179,203,151],[19,24930,24932],{"className":24931},[155,151],[19,24933,63],{"className":24934},[155,151],[19,24936,24937,24940],{"style":212},[19,24938],{"className":24939,"style":144},[143],[19,24941],{"className":24942,"style":220},[219,151],[19,24944,24945,24948],{"style":223},[19,24946],{"className":24947,"style":144},[143],[19,24949,24951],{"className":24950},[148,179,203,151],[19,24952,24954],{"className":24953},[155,151],[19,24955,402],{"className":24956},[155,151],[19,24958,242],{"className":24959},[241],[19,24961,24963],{"className":24962},[131],[19,24964,24966],{"className":24965,"style":249},[135],[19,24967],{},[19,24969],{"className":24970},[255,178,148,179,180],[19,24972,24973,24976],{"style":3968},[19,24974],{"className":24975,"style":144},[143],[19,24977],{"className":24978,"style":3975},[219],[19,24980,24981,24984],{"style":3978},[19,24982],{"className":24983,"style":144},[143],[19,24985,24987,24990],{"className":24986},[155],[19,24988,77],{"className":24989},[155,236],[19,24991,7425],{"className":24992},[255],[19,24994,242],{"className":24995},[241],[19,24997,24999],{"className":24998},[131],[19,25000,25002],{"className":25001,"style":19966},[135],[19,25003],{},[19,25005],{"className":25006},[255,178],[19,25008,25010,25013],{"className":25009},[155],[19,25011,15064],{"className":25012},[155,236],[19,25014,25016],{"className":25015},[122],[19,25017,25019],{"className":25018},[126],[19,25020,25022],{"className":25021},[131],[19,25023,25025],{"className":25024,"style":18443},[135],[19,25026,25027,25030],{"style":2136},[19,25028],{"className":25029,"style":298},[143],[19,25031,25033],{"className":25032},[148,149,150,151],[19,25034,77],{"className":25035},[155,236,151],[19,25037],{"className":25038,"style":607},[271],[19,25040,370],{"className":25041},[611],[19,25043],{"className":25044,"style":607},[271],[19,25046,25048,25051],{"className":25047},[103],[19,25049],{"className":25050,"style":4230},[107],[19,25052,18501],{"className":25053},[155,236],[15,25055,25056],{},"Then, we need to find this limit.",[15,25058,25059],{},"From the earlier Wallis formula:",[19,25061,25063],{"className":25062,"translate":23},[22],[19,25064,25066,25155],{"className":25065,"translate":23},[27],[19,25067,25069],{"className":25068},[31],[33,25070,25071],{"xmlns":35,"display":36},[38,25072,25073,25153],{},[41,25074,25075,25091,25145,25147],{},[10327,25076,25077,25083],{},[41,25078,25079,25081],{},[58,25080,10333],{},[47,25082,74],{},[41,25084,25085,25087,25089],{},[58,25086,4255],{},[47,25088,10342],{},[58,25090,10346],{"mathvariant":10345},[55,25092,25093,25111],{},[41,25094,25095,25097,25099,25101,25103,25105],{},[47,25096,412],{"stretchy":411},[51,25098,63],{},[58,25100,4255],{},[47,25102,422],{"stretchy":411},[47,25104,7425],{"stretchy":411},[65,25106,25107,25109],{},[47,25108,7425],{"stretchy":411},[51,25110,63],{},[41,25112,25113,25115,25117,25119,25121,25123,25125,25127,25129,25131,25133,25135,25137,25139,25141,25143],{},[47,25114,412],{"stretchy":411},[51,25116,63],{},[58,25118,4255],{},[47,25120,1066],{},[51,25122,402],{},[47,25124,422],{"stretchy":411},[47,25126,7425],{"stretchy":411},[47,25128,7425],{"stretchy":411},[47,25130,412],{"stretchy":411},[51,25132,63],{},[58,25134,4255],{},[47,25136,373],{},[51,25138,402],{},[47,25140,422],{"stretchy":411},[47,25142,7425],{"stretchy":411},[47,25144,7425],{"stretchy":411},[47,25146,370],{},[55,25148,25149,25151],{},[58,25150,60],{},[51,25152,63],{},[91,25154,14652],{"encoding":93},[19,25156,25158,25374],{"className":25157,"ariaHidden":99},[98],[19,25159,25161,25164,25217,25220,25365,25368,25371],{"className":25160},[103],[19,25162],{"className":25163,"style":12616},[107],[19,25165,25167],{"className":25166},[112,10645],[19,25168,25170,25209],{"className":25169},[126,127],[19,25171,25173,25206],{"className":25172},[131],[19,25174,25176,25196],{"className":25175,"style":4286},[135],[19,25177,25178,25181],{"style":10657},[19,25179],{"className":25180,"style":144},[143],[19,25182,25184],{"className":25183},[148,149,150,151],[19,25185,25187,25190,25193],{"className":25186},[155,151],[19,25188,4255],{"className":25189,"style":4293},[155,236,151],[19,25191,10342],{"className":25192},[611,151],[19,25194,10346],{"className":25195},[155,151],[19,25197,25198,25201],{"style":10678},[19,25199],{"className":25200,"style":144},[143],[19,25202,25203],{},[19,25204,10333],{"className":25205},[112],[19,25207,242],{"className":25208},[241],[19,25210,25212],{"className":25211},[131],[19,25213,25215],{"className":25214,"style":10696},[135],[19,25216],{},[19,25218],{"className":25219,"style":272},[271],[19,25221,25223,25226,25362],{"className":25222},[155],[19,25224],{"className":25225},[177,178],[19,25227,25229],{"className":25228},[55],[19,25230,25232,25354],{"className":25231},[126,127],[19,25233,25235,25351],{"className":25234},[131],[19,25236,25238,25294,25302],{"className":25237,"style":12635},[135],[19,25239,25240,25243],{"style":3956},[19,25241],{"className":25242,"style":144},[143],[19,25244,25246,25249,25252,25255,25258,25261,25264,25267,25270,25273,25276,25279,25282,25285,25288,25291],{"className":25245},[155],[19,25247,412],{"className":25248},[177],[19,25250,63],{"className":25251},[155],[19,25253,4255],{"className":25254,"style":4293},[155,236],[19,25256],{"className":25257,"style":920},[271],[19,25259,1066],{"className":25260},[788],[19,25262],{"className":25263,"style":920},[271],[19,25265,402],{"className":25266},[155],[19,25268,7889],{"className":25269},[255],[19,25271,412],{"className":25272},[177],[19,25274,63],{"className":25275},[155],[19,25277,4255],{"className":25278,"style":4293},[155,236],[19,25280],{"className":25281,"style":920},[271],[19,25283,373],{"className":25284},[788],[19,25286],{"className":25287,"style":920},[271],[19,25289,402],{"className":25290},[155],[19,25292,7889],{"className":25293},[255],[19,25295,25296,25299],{"style":3968},[19,25297],{"className":25298,"style":144},[143],[19,25300],{"className":25301,"style":3975},[219],[19,25303,25304,25307],{"style":3978},[19,25305],{"className":25306,"style":144},[143],[19,25308,25310,25313,25316,25319,25322],{"className":25309},[155],[19,25311,412],{"className":25312},[177],[19,25314,63],{"className":25315},[155],[19,25317,4255],{"className":25318,"style":4293},[155,236],[19,25320,12720],{"className":25321},[255],[19,25323,25325,25328],{"className":25324},[255],[19,25326,7425],{"className":25327},[255],[19,25329,25331],{"className":25330},[122],[19,25332,25334],{"className":25333},[126],[19,25335,25337],{"className":25336},[131],[19,25338,25340],{"className":25339,"style":12739},[135],[19,25341,25342,25345],{"style":12742},[19,25343],{"className":25344,"style":298},[143],[19,25346,25348],{"className":25347},[148,149,150,151],[19,25349,63],{"className":25350},[155,151],[19,25352,242],{"className":25353},[241],[19,25355,25357],{"className":25356},[131],[19,25358,25360],{"className":25359,"style":7939},[135],[19,25361],{},[19,25363],{"className":25364},[255,178],[19,25366],{"className":25367,"style":607},[271],[19,25369,370],{"className":25370},[611],[19,25372]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= 2n (2n-2) \\cdots 2 = 2^n n!",[19,25507,25509,25536,25566,25599],{"className":25508,"ariaHidden":99},[98],[19,25510,25512,25515,25518,25521,25524,25527,25530,25533],{"className":25511},[103],[19,25513],{"className":25514,"style":2167},[107],[19,25516,412],{"className":25517},[177],[19,25519,63],{"className":25520},[155],[19,25522,77],{"className":25523},[155,236],[19,25525,7889],{"className":25526},[255],[19,25528],{"className":25529,"style":607},[271],[19,25531,370],{"className":25532},[611],[19,25534],{"className":25535,"style":607},[271],[19,25537,25539,25542,25545,25548,25551,25554,25557,25560,25563],{"className":25538},[103],[19,25540],{"className":25541,"style":2167},[107],[19,25543,63],{"className":25544},[155],[19,25546,77],{"className":25547},[155,236],[19,25549,412],{"className":25550},[177],[19,25552,63],{"className":25553},[155],[19,25555,77],{"className":25556},[155,236],[19,25558],{"className":25559,"style":920},[271],[19,25561,373],{"className":25562},[788],[19,25564],{"className":25565,"style":920},[271],[19,25567,25569,25572,25575,25578,25581,25584,25587,25590,25593,25596],{"className":25568},[103],[19,25570],{"className":25571,"style":2167},[107],[19,25573,63],{"className":25574},[155],[19,25576,422],{"className":25577},[255],[19,25579],{"className":25580,"style":272},[271],[19,25582,4451],{"className":25583},[1353],[19,25585],{"className":25586,"style":272},[271],[19,25588,63],{"className":25589},[155],[19,25591],{"className":25592,"style":607},[271],[19,25594,370],{"className":25595},[611],[19,25597],{"className":25598,"style":607},[271],[19,25600,25602,25605,25634,25637],{"className":25601},[103],[19,25603],{"className":25604,"style":18443},[107],[19,25606,25608,25611],{"className":25607},[155],[19,25609,63],{"className":25610},[155],[19,25612,25614],{"className":25613},[122],[19,25615,25617],{"className":25616},[126],[19,25618,25620],{"className":25619},[131],[19,25621,25623],{"className":25622,"style":18443},[135],[19,25624,25625,25628],{"style":2136},[19,25626],{"className":25627,"style":298},[143],[19,25629,25631],{"className":25630},[148,149,150,151],[19,25632,77],{"className":25633},[155,236,151],[19,25635,77],{"className":25636},[155,236],[19,25638,7425],{"className":25639},[255],[19,25641,25643],{"className":25642,"translate":23},[22],[19,25644,25646,25732],{"className":25645,"translate":23},[27],[19,25647,25649],{"className":25648},[31],[33,25650,25651],{"xmlns":35,"display":36},[38,25652,25653,25729],{},[41,25654,25655,25657,25659,25661,25663,25665,25667,25669,25671,25673,25701,25703],{},[47,25656,412],{"stretchy":411},[51,25658,63],{},[58,25660,77],{},[47,25662,373],{},[51,25664,402],{},[47,25666,422],{"stretchy":411},[47,25668,7425],{"stretchy":411},[47,25670,7425],{"stretchy":411},[47,25672,370],{},[55,25674,25675,25687],{},[41,25676,25677,25679,25681,25683,25685],{},[47,25678,412],{"stretchy":411},[51,25680,63],{},[58,25682,77],{},[47,25684,422],{"stretchy":411},[47,25686,7425],{"stretchy":411},[41,25688,25689,25691,25693,25695,25697,25699],{},[47,25690,412],{"stretchy":411},[51,25692,63],{},[58,25694,77],{},[47,25696,422],{"stretchy":411},[47,25698,7425],{"stretchy":411},[47,25700,7425],{"stretchy":411},[47,25702,370],{},[55,25704,25705,25717],{},[41,25706,25707,25709,25711,25713,25715],{},[47,25708,412],{"stretchy":411},[51,25710,63],{},[58,25712,77],{},[47,25714,422],{"stretchy":411},[47,25716,7425],{"stretchy":411},[41,25718,25719,25725,25727],{},[65,25720,25721,25723],{},[51,25722,63],{},[58,25724,77],{},[58,25726,77],{},[47,25728,7425],{"stretchy":411},[91,25730,25731],{"encoding":93},"(2n-1)!! = \\frac{(2n)!}{(2n)!!} = \\frac{(2n)!}{2^n n!}",[19,25733,25735,25759,25780,25875],{"className":25734,"ariaHidden":99},[98],[19,25736,25738,25741,25744,25747,25750,25753,25756],{"className":25737},[103],[19,25739],{"className":25740,"style":2167},[107],[19,25742,412],{"className":25743},[177],[19,25745,63],{"className":25746},[155],[19,25748,77],{"className":25749},[155,236],[19,25751],{"className":25752,"style":920},[271],[19,25754,373],{"className":25755},[788],[19,25757],{"className":25758,"style":920},[271],[19,25760,25762,25765,25768,25771,25774,25777],{"className":25761},[103],[19,25763],{"className":25764,"style":2167},[107],[19,25766,402],{"className":25767},[155],[19,25769,7889],{"className":25770},[255],[19,25772],{"className":25773,"style":607},[271],[19,25775,370],{"className":25776},[611],[19,25778],{"className":25779,"style":607},[271],[19,25781,25783,25786,25866,25869,25872],{"className":25782},[103],[19,25784],{"className":25785,"style":7849},[107],[19,25787,25789,25792,25863],{"className":25788},[155],[19,25790],{"className":25791},[177,178],[19,25793,25795],{"className":25794},[55],[19,25796,25798,25855],{"className":25797},[126,127],[19,25799,25801,25852],{"className":25800},[131],[19,25802,25804,25824,25832],{"className":25803,"style":7868},[135],[19,25805,25806,25809],{"style":3956},[19,25807],{"className":25808,"style":144},[143],[19,25810,25812,25815,25818,25821],{"className":25811},[155],[19,25813,412],{"className":25814},[177],[19,25816,63],{"className":25817},[155],[19,25819,77],{"className":25820},[155,236],[19,25822,7889],{"className":25823},[255],[19,25825,25826,25829],{"style":3968},[19,25827],{"className":25828,"style":144},[143],[19,25830],{"className":25831,"style":3975},[219],[19,25833,25834,25837],{"style":3978},[19,25835],{"className":25836,"style":144},[143],[19,25838,25840,25843,25846,25849],{"className":25839},[155],[19,25841,412],{"className":25842},[177],[19,25844,63],{"className":25845},[155],[19,25847,77],{"className":25848},[155,236],[19,25850,12720],{"className":25851},[255],[19,25853,242],{"className":25854},[241],[19,25856,25858],{"className":25857},[131],[19,25859,25861],{"className":25860,"style":7939},[135],[19,25862],{},[19,25864],{"className":25865},[255,178],[19,25867],{"className":25868,"style":607},[271],[19,25870,370],{"className":25871},[611],[19,25873],{"className":25874,"style":607},[271],[19,25876,25878,25882],{"className":25877},[103],[19,25879],{"className":25880,"style":25881},[107],"height:2.113em;vertical-align:-0.686em;",[19,25883,25885,25888,25982],{"className":25884},[155],[19,25886],{"className":25887},[177,178],[19,25889,25891],{"className":25890},[55],[19,25892,25894,25974],{"className":25893},[126,127],[19,25895,25897,25971],{"className":25896},[131],[19,25898,25900,25943,25951],{"className":25899,"style":7868},[135],[19,25901,25902,25905],{"style":3956},[19,25903],{"className":25904,"style":144},[143],[19,25906,25908,25937,25940],{"className":25907},[155],[19,25909,25911,25914],{"className":25910},[155],[19,25912,63],{"className":25913},[155],[19,25915,25917],{"className":25916},[122],[19,25918,25920],{"className":25919},[126],[19,25921,25923],{"className":25922},[131],[19,25924,25926],{"className":25925,"style":15607},[135],[19,25927,25928,25931],{"style":12929},[19,25929],{"className":25930,"style":298},[143],[19,25932,25934],{"className":25933},[148,149,150,151],[19,25935,77],{"className":25936},[155,236,151],[19,25938,77],{"className":25939},[155,236],[19,25941,7425],{"className":25942},[255],[19,25944,25945,25948],{"style":3968},[19,25946],{"className":25947,"style":144},[143],[19,25949],{"className":25950,"style":3975},[219],[19,25952,25953,25956],{"style":3978},[19,25954],{"className":25955,"style":144},[143],[19,25957,25959,25962,25965,25968],{"className":25958},[155],[19,25960,412],{"className":25961},[177],[19,25963,63],{"className":25964},[155],[19,25966,77],{"className":25967},[155,236],[19,25969,12720],{"className":25970},[255],[19,25972,242],{"className":25973},[241],[19,25975,25977],{"className":25976},[131],[19,25978,25980],{"className":25979,"style":4009},[135],[19,25981],{},[19,25983],{"className":25984},[255,178],[19,25986,25988],{"className":25987,"translate":23},[22],[19,25989,25991,26103],{"className":25990,"translate":23},[27],[19,25992,25994],{"className":25993},[31],[33,25995,25996],{"xmlns":35,"display":36},[38,25997,25998,26100],{},[41,25999,26000,26002,26004,26006,26008,26010,26012,26014,26016,26018,26054,26056],{},[47,26001,412],{"stretchy":411},[51,26003,63],{},[58,26005,77],{},[47,26007,1066],{},[51,26009,402],{},[47,26011,422],{"stretchy":411},[47,26013,7425],{"stretchy":411},[47,26015,7425],{"stretchy":411},[47,26017,370],{},[55,26019,26020,26036],{},[41,26021,26022,26024,26026,26028,26030,26032,26034],{},[47,26023,412],{"stretchy":411},[51,26025,63],{},[58,26027,77],{},[47,26029,1066],{},[51,26031,63],{},[47,26033,422],{"stretchy":411},[47,26035,7425],{"stretchy":411},[41,26037,26038,26040,26042,26044,26046,26048,26050,26052],{},[47,26039,412],{"stretchy":411},[51,26041,63],{},[58,26043,77],{},[47,26045,1066],{},[51,26047,63],{},[47,26049,422],{"stretchy":411},[47,26051,7425],{"stretchy":411},[47,26053,7425],{"stretchy":411},[47,26055,370],{},[55,26057,26058,26074],{},[41,26059,26060,26062,26064,26066,26068,26070,26072],{},[47,26061,412],{"stretchy":411},[51,26063,63],{},[58,26065,77],{},[47,26067,1066],{},[51,26069,63],{},[47,26071,422],{"stretchy":411},[47,26073,7425],{"stretchy":411},[41,26075,26076,26088,26090,26092,26094,26096,26098],{},[65,26077,26078,26080],{},[51,26079,63],{},[41,26081,26082,26084,26086],{},[58,26083,77],{},[47,26085,1066],{},[51,26087,402],{},[47,26089,412],{"stretchy":411},[58,26091,77],{},[47,26093,1066],{},[51,26095,402],{},[47,26097,422],{"stretchy":411},[47,26099,7425],{"stretchy":411},[91,26101,26102],{"encoding":93},"(2n+1)!! = \\frac{(2n+2)!}{(2n+2)!!} = \\frac{(2n+2)!}{2^{n+1} (n+1)!}",[19,26104,26106,26130,26151,26270],{"className":26105,"ariaHidden":99},[98],[19,26107,26109,26112,26115,26118,26121,26124,26127],{"className":26108},[103],[19,26110],{"className":26111,"style":2167},[107],[19,26113,412],{"className":26114},[177],[19,26116,63],{"className":26117},[155],[19,26119,77],{"className":26120},[155,236],[19,26122],{"className":26123,"style":920},[271],[19,26125,1066],{"className":26126},[788],[19,26128],{"className":26129,"style":920},[271],[19,26131,26133,26136,26139,26142,26145,26148],{"className":26132},[103],[19,26134],{"className":26135,"style":2167},[107],[19,26137,402],{"className":26138},[155],[19,26140,7889],{"className":26141},[255],[19,26143],{"className":26144,"style":607},[271],[19,26146,370],{"className":26147},[611],[19,26149],{"className":26150,"style":607},[271],[19,26152,26154,26157,26261,26264,26267],{"className":26153},[103],[19,26155],{"className":26156,"style":7849},[107],[19,26158,26160,26163,26258],{"className":26159},[155],[19,26161],{"className":26162},[177,178],[19,26164,26166],{"className":26165},[55],[19,26167,26169,26250],{"className":26168},[126,127],[19,26170,26172,26247],{"className":26171},[131],[19,26173,26175,26207,26215],{"className":26174,"style":7868},[135],[19,26176,26177,26180],{"style":3956},[19,26178],{"className":26179,"style":144},[143],[19,26181,26183,26186,26189,26192,26195,26198,26201,26204],{"className":26182},[155],[19,26184,412],{"className":26185},[177],[19,26187,63],{"className":26188},[155],[19,26190,77],{"className":26191},[155,236],[19,26193],{"className":26194,"style":920},[271],[19,26196,1066],{"className":26197},[788],[19,26199],{"className":26200,"style":920},[271],[19,26202,63],{"className":26203},[155],[19,26205,7889],{"className":26206},[255],[19,26208,26209,26212],{"style":3968},[19,26210],{"className":26211,"style":144},[143],[19,26213],{"className":26214,"style":3975},[219],[19,26216,26217,26220],{"style":3978},[19,26218],{"className":26219,"style":144},[143],[19,26221,26223,26226,26229,26232,26235,26238,26241,26244],{"className":26222},[155],[19,26224,412],{"className":26225},[177],[19,26227,63],{"className":26228},[155],[19,26230,77],{"className":26231},[155,236],[19,26233],{"className":26234,"style":920},[271],[19,26236,1066],{"className":26237},[788],[19,26239],{"className":26240,"style":920},[271],[19,26242,63],{"className":26243},[155],[19,26245,12720],{"className":26246},[255],[19,26248,242],{"className":26249},[241],[19,26251,26253],{"className":26252},[131],[19,26254,26256],{"className":26255,"style":7939},[135],[19,26257],{},[19,26259],{"className":26260},[255,178],[19,26262],{"className":26263,"style":607},[271],[19,26265,370],{"className":26266},[611],[19,26268],{"className":26269,"style":607},[271],[19,26271,26273,26276],{"className":26272},[103],[19,26274],{"className":26275,"style":7849},[107],[19,26277,26279,26282,26412],{"className":26278},[155],[19,26280],{"className":26281},[177,178],[19,26283,26285],{"className":26284},[55],[19,26286,26288,26404],{"className":26287},[126,127],[19,26289,26291,26401],{"className":26290},[131],[19,26292,26294,26361,26369],{"className":26293,"style":7868},[135],[19,26295,26296,26299],{"style":3956},[19,26297],{"className":26298,"style":144},[143],[19,26300,26302,26340,26343,26346,26349,26352,26355,26358],{"className":26301},[155],[19,26303,26305,26308],{"className":26304},[155],[19,26306,63],{"className":26307},[155],[19,26309,26311],{"className":26310},[122],[19,26312,26314],{"className":26313},[126],[19,26315,26317],{"className":26316},[131],[19,26318,26320],{"className":26319,"style":12926},[135],[19,26321,26322,26325],{"style":12929},[19,26323],{"className":26324,"style":298},[143],[19,26326,26328],{"className":26327},[148,149,150,151],[19,26329,26331,26334,26337],{"className":26330},[155,151],[19,26332,77],{"className":26333},[155,236,151],[19,26335,1066],{"className":26336},[788,151],[19,26338,402],{"className":26339},[155,151],[19,26341,412],{"className":26342},[177],[19,26344,77],{"className":26345},[155,236],[19,26347],{"className":26348,"style":920},[271],[19,26350,1066],{"className":26351},[788],[19,26353],{"className":26354,"style":920},[271],[19,26356,402],{"className":26357},[155],[19,26359,12720],{"className":26360},[255],[19,26362,26363,26366],{"style":3968},[19,26364],{"className":26365,"style":144},[143],[19,26367],{"className":26368,"style":3975},[219],[19,26370,26371,26374],{"style":3978},[19,26372],{"className":26373,"style":144},[143],[19,26375,26377,26380,26383,26386,26389,26392,26395,26398],{"className":26376},[155],[19,26378,412],{"className":26379},[177],[19,26381,63],{"className":26382},[155],[19,26384,77],{"className":26385},[155,236],[19,26387],{"className":26388,"style":920},[271],[19,26390,1066],{"className":26391},[788],[19,26393],{"className":26394,"style":920},[271],[19,26396,63],{"className":26397},[155],[19,26399,12720],{"className":26400},[255],[19,26402,242],{"className":26403},[241],[19,26405,26407],{"className":26406},[131],[19,26408,26410],{"className":26409,"style":7939},[135],[19,26411],{},[19,26413],{"className":26414},[255,178],[15,26416,26417],{},"Thus, Wallis' formula becomes:",[19,26419,26421],{"className":26420,"translate":23},[22],[19,26422,26424,26521],{"className":26423,"translate":23},[27],[19,26425,26427],{"className":26426},[31],[33,26428,26429],{"xmlns":35,"display":36},[38,26430,26431,26518],{},[41,26432,26433,26449,26510,26512],{},[10327,26434,26435,26441],{},[41,26436,26437,26439],{},[58,26438,10333],{},[47,26440,74],{},[41,26442,26443,26445,26447],{},[58,26444,77],{},[47,26446,10342],{},[58,26448,10346],{"mathvariant":10345},[55,26450,26451,26484],{},[41,26452,26453,26460,26462,26464,26466,26472,26474,26476,26478,26480,26482],{},[65,26454,26455,26458],{},[51,26456,26457],{},"16",[58,26459,77],{},[47,26461,412],{"stretchy":411},[58,26463,77],{},[47,26465,7425],{"stretchy":411},[65,26467,26468,26470],{},[47,26469,422],{"stretchy":411},[51,26471,7450],{},[47,26473,412],{"stretchy":411},[51,26475,63],{},[58,26477,77],{},[47,26479,1066],{},[51,26481,63],{},[47,26483,422],{"stretchy":411},[41,26485,26486,26488,26490,26492,26494,26496,26498,26500,26502,26504,26506,26508],{},[47,26487,412],{"stretchy":411},[51,26489,63],{},[58,26491,77],{},[47,26493,422],{"stretchy":411},[47,26495,7425],{"stretchy":411},[47,26497,412],{"stretchy":411},[51,26499,63],{},[58,26501,77],{},[47,26503,1066],{},[51,26505,63],{},[47,26507,422],{"stretchy":411},[47,26509,7425],{"stretchy":411},[47,26511,370],{},[55,26513,26514,26516],{},[58,26515,60],{},[51,26517,63],{},[91,26519,26520],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{16^n (n!)^4 (2n+2)}{(2n)!(2n+2)!} = \\frac{\\pi}{2}",[19,26522,26524,26782],{"className":26523,"ariaHidden":99},[98],[19,26525,26527,26530,26583,26586,26773,26776,26779],{"className":26526},[103],[19,26528],{"className":26529,"style":12616},[107],[19,26531,26533],{"className":26532},[112,10645],[19,26534,26536,26575],{"className":26535},[126,127],[19,26537,26539,26572],{"className":26538},[131],[19,26540,26542,26562],{"className":26541,"style":4286},[135],[19,26543,26544,26547],{"style":15507},[19,26545],{"className":26546,"style":144},[143],[19,26548,26550],{"className":26549},[148,149,150,151],[19,26551,26553,26556,26559],{"className":26552},[155,151],[19,26554,77],{"className":26555},[155,236,151],[19,26557,10342],{"className":26558},[611,151],[19,26560,10346],{"className":26561},[155,151],[19,26563,26564,26567],{"style":10678},[19,26565],{"className":26566,"style":144},[143],[19,26568,26569],{},[19,26570,10333],{"className":26571},[112],[19,26573,242],{"className":26574},[241],[19,26576,26578],{"className":26577},[131],[19,26579,26581],{"className":26580,"style":15545},[135],[19,26582],{},[19,26584],{"className":26585,"style":272},[271],[19,26587,26589,26592,26770],{"className":26588},[155],[19,26590],{"className":26591},[177,178],[19,26593,26595],{"className":26594},[55],[19,26596,26598,26762],{"className":26597},[126,127],[19,26599,26601,26759],{"className":26600},[131],[19,26602,26604,26648,26656],{"className":26603,"style":12635},[135],[19,26605,26606,26609],{"style":3956},[19,26607],{"className":26608,"style":144},[143],[19,26610,26612,26615,26618,26621,26624,26627,26630,26633,26636,26639,26642,26645],{"className":26611},[155],[19,26613,412],{"className":26614},[177],[19,26616,63],{"className":26617},[155],[19,26619,77],{"className":26620},[155,236],[19,26622,12720],{"className":26623},[255],[19,26625,412],{"className":26626},[177],[19,26628,63],{"className":26629},[155],[19,26631,77],{"className":26632},[155,236],[19,26634],{"className":26635,"style":920},[271],[19,26637,1066],{"className":26638},[788],[19,26640],{"className":26641,"style":920},[271],[19,26643,63],{"className":26644},[155],[19,26646,12720],{"className":26647},[255],[19,26649,26650,26653],{"style":3968},[19,26651],{"className":26652,"style":144},[143],[19,26654],{"className":26655,"style":3975},[219],[19,26657,26658,26661],{"style":3978},[19,26659],{"className":26660,"style":144},[143],[19,26662,26664,26667,26697,26700,26703,26706,26735,26738,26741,26744,26747,26750,26753,26756],{"className":26663},[155],[19,26665,402],{"className":26666},[155],[19,26668,26670,26674],{"className":26669},[155],[19,26671,26673],{"className":26672},[155],"6",[19,26675,26677],{"className":26676},[122],[19,26678,26680],{"className":26679},[126],[19,26681,26683],{"className":26682},[131],[19,26684,26686],{"className":26685,"style":15017},[135],[19,26687,26688,26691],{"style":12742},[19,26689],{"className":26690,"style":298},[143],[19,26692,26694],{"className":26693},[148,149,150,151],[19,26695,77],{"className":26696},[155,236,151],[19,26698,412],{"className":26699},[177],[19,26701,77],{"className":26702},[155,236],[19,26704,7425],{"className":26705},[255],[19,26707,26709,26712],{"className":26708},[255],[19,26710,422],{"className":26711},[255],[19,26713,26715],{"className":26714},[122],[19,26716,26718],{"className":26717},[126],[19,26719,26721],{"className":26720},[131],[19,26722,26724],{"className":26723,"style":12739},[135],[19,26725,26726,26729],{"style":12742},[19,26727],{"className":26728,"style":298},[143],[19,26730,26732],{"className":26731},[148,149,150,151],[19,26733,7450],{"className":26734},[155,151],[19,26736,412],{"className":26737},[177],[19,26739,63],{"className":26740},[155],[19,26742,77],{"className":26743},[155,236],[19,26745],{"className":26746,"style":920},[271],[19,26748,1066],{"className":26749},[788],[19,26751],{"className":26752,"style":920},[271],[19,26754,63],{"className":26755},[155],[19,26757,422],{"className":26758},[255],[19,26760,242],{"className":26761},[241],[19,26763,26765],{"className":26764},[131],[19,26766,26768],{"className":26767,"style":7939},[135],[19,26769],{},[19,26771],{"className":26772},[255,178],[19,26774],{"className":26775,"style":607},[271],[19,26777,370],{"className":26778},[611],[19,26780],{"className":26781,"style":607},[271],[19,26783,26785,26788],{"className":26784},[103],[19,26786],{"className":26787,"style":5766},[107],[19,26789,26791,26794,26847],{"className":26790},[155],[19,26792],{"className":26793},[177,178],[19,26795,26797],{"className":26796},[55],[19,26798,26800,26839],{"className":26799},[126,127],[19,26801,26803,26836],{"className":26802},[131],[19,26804,26806,26817,26825],{"className":26805,"style":5785},[135],[19,26807,26808,26811],{"style":3956},[19,26809],{"className":26810,"style":144},[143],[19,26812,26814],{"className":26813},[155],[19,26815,63],{"className":26816},[155],[19,26818,26819,26822],{"style":3968},[19,26820],{"className":26821,"style":144},[143],[19,26823],{"className":26824,"style":3975},[219],[19,26826,2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\\sim a e^{-n} n^{n + \\frac{1}{2}}",[19,26906,26908,26929],{"className":26907,"ariaHidden":99},[98],[19,26909,26911,26914,26917,26920,26923,26926],{"className":26910},[103],[19,26912],{"className":26913,"style":4286},[107],[19,26915,77],{"className":26916},[155,236],[19,26918,7425],{"className":26919},[255],[19,26921],{"className":26922,"style":607},[271],[19,26924,17990],{"className":26925},[611],[19,26927],{"className":26928,"style":607},[271],[19,26930,26932,26936,26939,26975],{"className":26931},[103],[19,26933],{"className":26934,"style":26935},[107],"height:1.004em;",[19,26937,18501],{"className":26938},[155,236],[19,26940,26942,26945],{"className":26941},[155],[19,26943,15064],{"className":26944},[155,236],[19,26946,26948],{"className":26947},[122],[19,26949,26951],{"className":26950},[126],[19,26952,26954],{"className":26953},[131],[19,26955,26958],{"className":26956,"style":26957},[135],"height:0.8213em;",[19,26959,26960,26963],{"style":2136},[19,26961],{"className":26962,"style":298},[143],[19,26964,26966],{"className":26965},[148,149,150,151],[19,26967,26969,26972],{"className":26968},[155,151],[19,26970,373],{"className":26971},[155,151],[19,26973,77],{"className":26974},[155,236,151],[19,26976,26978,26981],{"className":26977},[155],[19,26979,77],{"className":26980},[155,236],[19,26982,26984],{"className":26983},[122],[19,26985,26987],{"className":26986},[126],[19,26988,26990],{"className":26989},[131],[19,26991,26993],{"className":26992,"style":26935},[135],[19,26994,26995,26998],{"style":16311},[19,26996],{"className":26997,"style":144},[143],[19,26999,27001],{"className":27000},[148,149,150,151],[19,27002,27004,27007,27010],{"className":27003},[155,151],[19,27005,77],{"className":27006},[155,236,151],[19,27008,1066],{"className":27009},[788,151],[19,27011,27013,27016,27075],{"className":27012},[155,151],[19,27014],{"className":27015},[177,178,148,179,180],[19,27017,27019],{"className":27018},[55],[19,27020,27022,27067],{"className":27021},[126,127],[19,27023,27025,27064],{"className":27024},[131],[19,27026,27028,27042,27050],{"className":27027,"style":15704},[135],[19,27029,27030,27033],{"style":196},[19,27031],{"className":27032,"style":144},[143],[19,27034,27036],{"className":27035},[148,179,203,151],[19,27037,27039],{"className":27038},[155,151],[19,27040,63],{"className":27041},[155,151],[19,27043,27044,27047],{"style":212},[19,27045],{"className":27046,"style":144},[143],[19,27048],{"className":27049,"style":220},[219,151],[19,27051,27052,27055],{"style":223},[19,27053],{"className":27054,"style":144},[143],[19,27056,27058],{"className":27057},[148,179,203,151],[19,27059,27061],{"className":27060},[155,151],[19,27062,402],{"className":27063},[155,151],[19,27065,242],{"className":27066},[241],[19,27068,27070],{"className":27069},[131],[19,27071,27073],{"className":27072,"style":249},[135],[19,27074],{},[19,27076],{"className":27077},[255,178,148,179,180],[15,27079,27080],{},"Substitute into:",[19,27082,27084],{"className":27083,"translate":23},[22],[19,27085,27087,27177],{"className":27086,"translate":23},[27],[19,27088,27090],{"className":27089},[31],[33,27091,27092],{"xmlns":35,"display":36},[38,27093,27094,27174],{},[41,27095,27096,27112,27166,27168],{},[10327,27097,27098,27104],{},[41,27099,27100,27102],{},[58,27101,10333],{},[47,27103,74],{},[41,27105,27106,27108,27110],{},[58,27107,77],{},[47,27109,10342],{},[58,27111,10346],{"mathvariant":10345},[55,27113,27114,27128],{},[41,27115,27116,27122],{},[65,27117,27118,27120],{},[58,27119,18501],{},[51,27121,63],{},[65,27123,27124,27126],{},[58,27125,15064],{},[51,27127,63],{},[41,27129,27130,27132],{},[51,27131,7450],{},[65,27133,27134,27152],{},[41,27135,27136,27138,27150],{},[47,27137,412],{"fence":99},[55,27139,27140,27148],{},[41,27141,27142,27144,27146],{},[58,27143,77],{},[47,27145,1066],{},[51,27147,402],{},[58,27149,77],{},[47,27151,422],{"fence":99},[41,27153,27154,27156,27158,27160],{},[51,27155,63],{},[58,27157,77],{},[47,27159,1066],{},[55,27161,27162,27164],{},[51,27163,4438],{},[51,27165,63],{},[47,27167,370],{},[55,27169,27170,27172],{},[58,27171,60],{},[51,27173,63],{},[91,27175,27176],{"encoding":93},"\\lim_{n \\rightarrow \\infty} \\frac{a^2 e^2}{4 \\left(\\frac{n+1}{n}\\right)^{2n + \\frac{3}{2}}} = 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formula finds a relationship of n!,nⁿ,eⁿ while n trends to infinity.",[28317],"mathematics","Df-8ho11rFhshSb-Sfb-zryDiVQlLv2QuB1tk9P53fo",1777090294956]