1.3二项式定理(通用)VIP免费

Howwouldyoufind?1011.011.0221)01.01(1.00521)20.01(1.0151.0321)30.01(1.011.021.0421)40.01(1.0251.0521)50.01(xxx211)1(121Review--BinomialExpansionrnxrnxnxnxnx3232111NnNisanaturalnumber---anypositiveintegerorzerornxrrnnnnxnnnxnnnxx!12)1(!32)1(!2)1(1132KWhetherncanbeaREALNUMBER？TheGeneralBiomialExpansion广义二项式定理2018-6-15陈含烨theGeneralBinomialExpansionrnxrrnnnnxnnnxnnnxx!12)1(!32)1(!2)1(1132RnIt'struewhennisanyrealnumber,buttherearetwoimportantdifferencestonotewhenNnProvingthisresultisbeyondthescopeofanA-levelcousebutyoucanassumethatitistrueTheseriesisinfiniteTheexpansionofisvalidonlyifnx11x14641413313121211110,!4321,!32)1(,!21,,1nnnnnnnnnnnnnnn0000…(infinitelymanyzero)000…(infinitelymanyzero)00…(infinitelymanyzero)0…(infinitelymanyzero)…(infinitelymanyzero)HowtounderstandtheInfinite？•Considerthecoefficientsinthebinomialexpansion:Ofcourse,it'susualtodiscardallthezeros.Findthecoefficientsinthebinomialexpansion3n!46543-!3543-!2433-1gives1510-63-1isthat21n!425232121!3232121!22121211gives1285-16181-211isthat432314106311xxxxxthatso432211285161812111xxxxxthatso1xWhy&approximation1xTheBinomialTheoremstatesthatforanyvalueofn:where•if,xmaytakeanyvalue•if,32!32)1(!2)1(11xnnnxnnnxxnNnNn1xEXAMPLE7.1•Expand(1-x)-2asasriesofascendingpowersofxuptoandincludingtheterminx3,statingthesetofvaluesofxforwhichtheexpansionisvalid.1when4321)1(322xxxxxEXAMPLE7.2•Findaquadraticapproximationforandstateforwhichvaluesofttheexpansionisvalid.t2112123121221twhentttHowwouldyoufind?101?110010121nnnaxaxa132!321!211axnnnaxnnaxnan1axfor221212101.0!2212101.02111001.011001100101SoEXAMPLE7.3•Expand(2+x)-3asasriesofascendingpowersofxuptoandincludingtheterminx2,statingthesetofvaluesofxforwhichtheexpansionisvalid.23332!243231812122xxxx2163163812xwhenxxSummaryTheBinomialTheoremif,xmaytakeanyvalueif,32!32)1(!2)1(11xnnnxnnnxxnNnNn1xnnnaxaxa1121axfor3fromparticulartogeneral，observation-induction-analogy-speculation-certification从特殊到一般，观察-归纳-类比-猜想-证明•Around1665,IsaacNewtongeneralizedthebinomialtheoremtoallowrealexponentsotherthannonnegativeintegers.Inthisgeneralization,thefinitesumisreplacedbyaninfiniteseries.•http://open.163.com/movie/2011/10/L/F/M8HH84LV3_M8HOR7DLF.htmlHistoryThankYou!p1636,7-22132