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1、PowerSeriesExpansionandItsApplicationsIntheprevioussection,wediscusstheconvergenceofpowerseries,initsconvergenceregion,thepowerseriesalwaysconvergestoafunction.Forthesimplepowerseries,butalsowithitemizedderivative,orquadraturemethods,findthisandfunction
2、.Thissectionwilldiscussanotherissue,foranarbitraryfunction,canbeexpandedinapowerseries,andlaunchedinto.Whetherthepowerseriesasandfunction?Thefollowingdiscussionwilladdressthisissue.1Maclaurin(Maclaurin)formulaPolynomialpowerseriescanbeseenasanextensiono
3、freality,soconsiderthefunctioncanexpandintopowerseries,youcanfromthefunctionandpolynomialsstarttosolvethisproblem.Tothisend,togiveherewithoutproofthefollowingformula.Taylor(Taylor)formula,ifthefunctionatinaneighborhoodthatuntilthederivativeoforder,theni
4、ntheneighborhoodofthefollowingformula:(9-5-1)AmongThatfortheLagrangianremainder.That(9-5-1)-typeformulafortheTaylor.Ifso,get,(9-5-2)Atthispoint,().That(9-5-2)typeformulafortheMaclaurin.Formulashowsthatanyfunctionaslongasuntilthederivative,canbeequaltoap
5、olynomialandaremainder.Wecallthefollowingpowerseries(9-5-3)FortheMaclaurinseries.So,isittofortheSumfunctions?IftheorderMaclaurinseries(9-5-3)thefirstitemsandfor,whichThen,theseries(9-5-3)convergestothefunctiontheconditions.NotingMaclaurinformula(9-5-2)a
6、ndtheMaclaurinseries(9-5-3)therelationshipbetweentheknownThus,whenThere,Viceversa.Thatif,Unitsmust.ThisindicatesthattheMaclaurinseries(9-5-3)toandfunctionastheMaclaurinformula(9-5-2)oftheremainderterm(when).Inthisway,wegetafunctionthepowerseriesexpansio
7、n:.(9-5-4)Itisthefunctionthepowerseriesexpression,if,thefunctionofthepowerseriesexpansionisunique.Infact,assumingthefunctionf(x)canbeexpressedaspowerseries,(9-5-5)Well,accordingtotheconvergenceofpowerseriescanbeitemizedwithinthenatureofderivation,andthe
8、nmake(powerseriesapparentlyconvergesinthepoint),itiseasytoget.Substitutingtheminto(9-5-5)type,incomeandtheMaclaurinexpansionof(9-5-4)identical.Insummary,ifthefunctionf(x)containszeroinarangeofarbitraryorderderivative,andinthisran
