函数幂级数毕业论文中英文资料外文翻译文献.docx
PowerSeriesExpansionandItsApp1.icationsIntheprevioussection,wediscusstheconvergenceofpowerseries,initsconvergenceregion,thepowerseriesa1.waysconvergestoafunction.Foi-thesimp1.epowerseries,buta1.sowithitemizedderivative,orquadraturemethods,findthisandfunction.Thissectionwi1.1.discussanotherissue,foranarbitraryfunction/(x),canbeexpandedinapowerseries,and1.aunchedinto.Whetherthepowerseries/(x)asandfunction?Thefo1.1.owingdiscussionwi1.1.addressthisissue.1Mac1.aurin(Mac1.aurin)formu1.aPo1.ynomia1.powerseriescanbeseenasanextensionofrea1.ity,soconsiderthefunction/（八）canexpandintopowerseries,youcanfromthefunction/(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)(9-5-1)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.owingpowerseries/(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.ationshipbetweentheknownThus,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,thefunctionofthepowerseriesexpansionisunique.In!"act,assumingthefunction/V)canbeexpressedaspowerseries/(*)=ZaI1.x"=<tt+aix+aix2+'''+uxm+'',(9-5-5)We1.1.,accordingtotheconvergenceofpowerseriescanbeitemizedwithinthenatureofderivation,andthenmakeX=O(powerseriesapparent1.yconvergesinthex=0point),itiseasytogetSubstitutingtheminto(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.PowerSeriesKnownastheTay1.orseries.Second,primaryfunctionofpowerseriesexpansionMac1.aurinformu1.ausingthefunction/()expandedinpowerseriesmethod,ca1.1.edthedirectexpansionmethod.Examp1.e1Testthefunction/(x)=<,aexpandedinpowerseriesofx.So1.utionbecauseThereforeSowegetthepowerseries1.+x+-.r+x+,(9-5-6)2!n,.Obvious1y,(9-5-6)typeconvergenceinterva1.(-o,÷x>),As(9-5-6)whethertypef(x)=e'isSumfunction,thatis,whetheritconvergestof(x)=e".buta1.soexamineremainder(.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.aareexpandedinpowerseriesmethod,a1.thoughtheprocedureisc1.ear,butoperatorsareoftentooCumbersome,soitisgenera1.1.ymoreconvenienttousethefo1.1.owingpowerseriesexpansionmethod.Priortothis,Wehavebeenafunction,e'andSinXpowerseries1.-.rexpansion,theuseoftheseknownexpansionbypowerseriesofoperations,Wecanachievemanyfunctionsofpowerseriesexpansion.Thisdemandfunctionofpowerseriesexpansionmethodisca1.1.edindirectexpansion.Example 2Findthefunctionf(x)=cosx,.r=0,Departmentinthepowerseriesexpansion.So1.utionbecauseAndTherefore,thepowerseriescanbeitemizedaccordingtotheru1.esofderivationcanbeThird,thefunctionpowerseriesexpansionoftheapp1.icationexamp1.eTheapp1.icationofpowerseriesexpansionisextensive,forexamp1.e,canuseittosetsomenumerica1.Orotherapproximateca1.cu1.ationofintegra1.va1.ue.Example 3 UsingtheexpansiontoCStimatearc1.anXtheva1.ueof11.So1.utionbecausearctanI=-4BecauseofSothereAvaiIab1.erightendofthefirstnitemsoftheseriesandasanapproximationof11.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(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，
