1、PowerSeriesExpansionandItsApp1.icationsIntheprevioussection,wediscusstheconvergenceofpowerseries,initsconvergenceregion,thepowerseriesa1.waysconvergestoafunction.Foi-thesimp1.epowerseries,buta1.sowithitemizedderivative,orquadraturemethods,findthisandfunction.Thissectionwi1.1.discussanotherissue,fora
2、narbitraryfunction/(x),canbeexpandedinapowerseries,and1.aunchedinto.Whetherthepowerseries/(x)asandfunction?Thefo1.1.owingdiscussionwi1.1.addressthisissue.1Mac1.aurin(Mac1.aurin)formu1.aPo1.ynomia1.powerseriescanbeseenasanextensionofrea1.ity,soconsiderthefunction/（八）canexpandintopowerseries,youcanfro
3、mthefunction/(x)andpo1.ynomia1.sstarttoso1.vethisprob1.em.Tothisend,togiveherewithoutproofthefo1.1.owingformu1.a.Tay1.or(Tay1.or)formu1.a,ifthefunctionf(x)at.x=xvinaneighborhoodthatunti1thederivativeofordern+1,thenintheneighborhoodofthefo1.1.owingformu1.a:f(x)=f(-vn)+(axn)+(a-x0)+-+(.r-x0)11+(x)(9-5
4、1)AmongThatr(.v)forthe1.agrangianremainder.That(9-5-1)-typeformu1.afortheTay1.or.Ifso=0,get(9-5-2)/(X)=/(O)+x+x2+.r+,(x),Atthispoint,That(9-5-2)typeformu1.afortheMac1.aurin.Formu1.ashowsthatanyfunctionf(x)as1.ongasunti1.the/+1derivative,ncanbeequa1.toapo1.ynomia1.andaremainder.Weca1.1.thefo1.1.owin
5、gpowerseries/(x)=/(0)+,(0).r+x2+-+(9-5-3)2!FortheMac1.aurinseries.So,isittof()fortheSumfunctions?IftheorderMac1.aurinseries(9-5-3)thefirstj+iternsandforS.i(x)fwhichThen,theseries(9-5-3)convergestothefunction/(x)theconditionsNotingMac1.aurinformu1.a(9-5-2)andtheMac1.aurinseries(9-5-3)there1.ationship
6、betweentheknownThus,whenThere,Viceversa.ThatifInitsmustThisindicatesthattheMac1.aurinseries(9-5-3)tof(xandfunctionastheMac1.aurinfo11nu1.a(9-5-2)oftheremaindertermrn(x)0(when).Inthisway,wegetafunctionf()thepowerseriesexpansion:=x=/(0)+,(0)x+.r+.(9-5-4)然！Itisthefundion/(x)thepowerseriesexpression,if,
7、thefunctionofthepowerseriesexpansionisunique.In!act,assumingthefunction/V)canbeexpressedaspowerseries/(*)=ZaI1.x=tt+aix+aix2+uxm+,(9-5-5)We1.1.,accordingtotheconvergenceofpowerseriescanbeitemizedwithinthenatureofderivation,andthenmakeX=O(powerseriesapparent1.yconvergesinthex=0point),itiseasytogetSub
8、stitutingtheminto(9-55)type,incomeandf(x)theMac1.aurinexpansionof(9-5-4)identica1.Insummary,ifthefunctionf(x)containszeroinarangeofarbitraryorderderivative,andinthisrangeofMac1.aurinformu1.aintheremaindertozeroasthe1imit(whenn-*,),then,thefunctionf(x)canstartfomingas(9-5-4)typeofpowerseries.PowerSer
9、iesKnownastheTay1.orseries.Second,primaryfunctionofpowerseriesexpansionMac1.aurinformu1.ausingthefunction/()expandedinpowerseriesmethod,ca1.1.edthedirectexpansionmethod.Examp1.e1Testthefunction/(x)=),As(9-5-6)whethertypef(x)=eisSumfunction,thatis,whetheritconvergestof(x)=e.buta1.soexamineremainder(.
10、r).Because,三-11(且q-w，U?十U-ThereforeNotingtheva1.ueofanysetx,/isafixedconstant,whi1.etheseries(9-5-6)isabso1.ute1.yconvergent,sothegenera1.whentheitemwhenn,Hut,u-0,sowhenn-*,5+)!thereFromthisThisindicatesthattheseries(9-5-6)doesconvergeto/(x)=e,thereforeSuchuseofMac1.aurinformu1.aareexpandedinpowerse
11、riesmethod,a1.thoughtheprocedureisc1.ear,butoperatorsareoftentooCumbersome,soitisgenera1.1.ymoreconvenienttousethefo1.1.owingpowerseriesexpansionmethod.Priortothis,Wehavebeenafunction,eandSinXpowerseries1.-.rexpansion,theuseoftheseknownexpansionbypowerseriesofoperations,Wecanachievemanyfunctionsofpo
12、werseriesexpansion.Thisdemandfunctionofpowerseriesexpansionmethodisca1.1.edindirectexpansion.Example 2Findthefunctionf(x)=cosx,.r=0,Departmentinthepowerseriesexpansion.So1.utionbecauseAndTherefore,thepowerseriescanbeitemizedaccordingtotheru1.esofderivationcanbeThird,thefunctionpowerseriesexpansionof
13、theapp1.icationexamp1.eTheapp1.icationofpowerseriesexpansionisextensive,forexamp1.e,canuseittosetsomenumerica1.Orotherapproximateca1.cu1.ationofintegra1.va1.ue.Example 3 UsingtheexpansiontoCStimatearc1.anXtheva1.ueof11.So1.utionbecausearctanI=-4BecauseofSothereAvaiIab1.erightendofthefirstnitemsofthe
14、seriesandasanapproximationof11.However,theconvergenceisverys1.owprogressiontogetenoughiternstogetmoreaccurateestimatesof11va1.ue.此外文文献选白J-:Wa1.ter.Rudin.数学分析原理(英文版)M北京：机械工业出版社.嘉级数的键开与其应用在上一节中，我们探讨了寨级数的收敛性，在其收敛域内，寨级数总是收敛r一个和函数.对r一些简洁的耗级数，还可以借助逐项求导或求积分的方法，求出这个和函数.本节将要探讨另外一个问题，对r随意一个函数人幻，能否将其绽开成一个帮级数，以
15、与绽开成的寨级数是否以八幻为和函数？下面的探讨将解决这一问题.一、马克劳林(MaCIaUrin)公式品级数事实上可以视为多项式的延长，因此在考虑函数/(X)能否绽开成品级数时，可以从函数八)与多项式的关系入手来解决这个问题.为此，这里不加证明地给出如下的公式.泰勒(TayIor)公式假如函数人工)在X=M的某一邻域内，有直到+1阶的导数，则在这个邻域内有如下公式：/(X)=/)+/(.%Xx-AU)+，：)(XTO-+/J)*-为)+1(X)，(952;！1)其中称MX)为拉格朗日型余项.称(951)式为泰勒公式.假如令Xo=0,就得到/(x)=(0)+x+-+.,+(),(952)此时，称(
16、952)式为马克劳林公式.公式说明，任一函数八幻只要有直到+1阶导数，就可等于某个次多项式与一个余项的和.我们称下列耗级数/(X)=/(O)+/W+.V2+-+Z+-(953)2!/!为马克劳林级数.那么，它是否以X)为和函数呢?若令马克劳林级数(953)的前+1项和为5幻，即那么，级数(953)收敛于函数外的条件为留意到马克劳林公式(952)与马克劳林级数(953)的关系，可知于是，当时，有反之亦然.即若则必有这表明，马克劳林级数(953)以/(*)为和函数。马克劳林公式(952)中的余项4,a)-0(当”T8时).这样，我们就得到了函数“)的恭级数绽开式：/工/WOHr(O)A四八+且F+
17、95M加2!w!4)它就是函数的第级数表达式，也就是说，函数的塞级数绽开式是唯一的.事实上，假设函数X)可以表示为品级数/(x)=Vn=W1.+ax+arr+a”+,那么，依据品级数在收敛域内可逐项求导的性质，再令X=O(耗级数明显在X=O点收敛)，就简洁得到将它们代入(955)式，所得与/)的马克劳林绽开式(954)完全相同.综上所述，假如函数刈在包含零的某区间内有随意阶导数，旦在此区间内的马克劳林公式中的余项以零为极限(当T=O时)，那么,函数f()就可绽开成形如(954)式的塞级数.耗级数称为泰勒级数.二、初等函数的耗级数绽开式利用马克劳林公式将函数人工)绽开成耗级数的方法，称为干脆绽开法.例1试将函数/(x)=e绽开成X的察级数.解因为所以于是我们得到常级数1.+x+-X2+,+x+(956)2!！明显，(956)式的收敛区间为(f),至r(956)式是否以/(x)=e*为和函数，即它是否收敛于/(x)=e*,还要考察余项q(x).因为(04又因所以有可用右端级数的前项之和作为”的近似值.但由于级数收敛的速度特别慢，要取足够多的项才能得到11的较精确的估计值.此外文文献选自于：Wa1.ter.Rdin.数学分析原理（英文版HM.北京：机械工业出版社.
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