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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)Among(x)=(-.vn,That()forthe1.agrangianremainder.That(9-5-1)-typeformu1.afortheTay1.or.Ifso=0,get/(.v)=(0)+.r+1+.r+r,(),(9-5-2)tthispoint,4.(x)=厂V*)I(m+1)!(M+1.)!Tha1.(9-5-2)typefor11u1.afortheMac1.aurin.Formu1.ashowsthatanyfundion/(.v)as1.ongasunti1the11+1.derivative,ncanbeequa1.toapo1.ynomia1.an
5、daremainder.Weca1.1.thefo1.1.owingpowerseries/(-)=/(0)+,(0)x+.+1.().ro+2!j!(9-5-3)FortheMac1.aurinseries.So,isittof(x)fortheSumfunctions?IftheorderMac1.aurinseries(9-5-3)thefirst+1.iternsandforS11.(x),which5,i(x)=/(0)+,()x+零+-+Z2!！Then,theseries(9-5-3)convergestothefunctionf(x)theconditionsIimSnTa)=
6、(x).NotingMac1.aurinformu1.a(9-5-2)andtheMac1.aurinseries(9-5-3)there1.ationshipbetweentheknown/(K)=SII“(x)U)Thus,when,+),As(9-5-6)whethertypef(x=eisSumfunction,thatis,whetheritconvergestof(x)=eA,buta1.soexamineremainderrn(x).Becauseb(oe),且e,4同IM1.M，U?十1.)Thereforek=-1.r,sothegenera1.whentheitcmwhen
7、noo,sowhenn-*,there(11+1)!0,Fromthis1.imq(x)=0Thisindicatesthattheseries(9-5-6)doesconvergeto/(x)=er,thereforee1.=1+-x2+,+,+,(o.r+oo).2!nSuchuseofMac1aurinformu1.aareexpandedinpowerseriesmethod,a1.thoughtheprocedureisc1.ear,butoperatorsareoftentooCumbersome,soitisgenera1.1.ymoreconvenienttousethefo1
8、.1.owingpowerseriesexpansionmethod.Priortothis,Wehavebeenafunctione,andsin.rpowerseries-xexpansion,theuseoftheseknownexpansionbypowerseriesofoperations,wecanachievemanyfunctionsofpowerseriesexpansion.Thisdemandfunctionofpowerseriesexpansionmethodisca1.1.edindirectexpansion.Example 2becauseFindthefun
9、ction/(x)=cos,x=0,Departmentinthepowerseriesexpansion.(SinX)=cos.t,So1.utionsin.v=And3!5!(2h+I)!Therefore,thepowerseriescanbeitemizedaccordingtotheru1.esofderivationcanbecos=1-+-A4+(-1.)!x1.11+,(-co.v+x)2!4!(2n)!Third,thefunctionpowerseriesexpansionoftheapp1.icationexamp1.eTheapp1.icationofpowerseri
10、esexpansionisextensive,forexamp1.e,canuseittosetsomenumerica1.orotherapproximateca1.cu1.ationofintegra1.va1ue.Example 3 Usingtheexpansiontoestimatearciaii.vtheva1.ueof11.So1.utionbecausearctan1=-4Becauseofarctana,=A-+-+,(-1x1.),357SothereAvaiIab1.erightendofthefirstnitemsoftheseriesandasanapproximationof11.However,theconve
