函数幂级数毕业论文中英文资料外文翻译文献.docx

PowerSeriesExpansionandItsApplications

Intheprevioussection,wediscusstheconvergenceofpowerseries,initsconvergenceregion,thepowerseriesalwaysconvergestoafunction.Forthesimplepowerseries,butalsowithitemizedderivative,orquadraturemethods,findthisandfunction.Thissectionwilldiscussanotherissue,foranarbitraryfunction,canbeexpandedinapowerseries,andlaunchedinto.

Whetherthepowerseriesasandfunction?Thefollowingdiscussionwilladdressthisissue.

1Maclaurin(Maclaurin)formula

Polynomialpowerseriescanbeseenasanextensionofreality,soconsiderthefunctioncanexpandintopowerseries,youcanfromthefunctionandpolynomialsstarttosolvethisproblem.Tothisend,togiveherewithoutproofthefollowingformula.

Taylor(Taylor)formula,ifthefunctionatinaneighborhoodthatuntilthederivativeoforder,thenintheneighborhoodofthefollowingformula:

(9-5-1)

Among

ThatfortheLagrangianremainder.That(9-5-1)-typeformulafortheTaylor.

Ifso,get

,(9-5-2)

Atthispoint,

That(9-5-2)typeformulafortheMaclaurin.

Formulashowsthatanyfunctionaslongasuntilthederivative,canbeequaltoapolynomialandaremainder.

Wecallthefollowingpowerseries

(9-5-3)

FortheMaclaurinseries.

So,isittofortheSumfunctions?IftheorderMaclaurinseries(9-5-3)thefirstitemsandfor,which

Then,theseries(9-5-3)convergestothefunctiontheconditions

NotingMaclaurinformula(9-5-2)andtheMaclaurinseries(9-5-3)therelationshipbetweentheknown

Thus,when

There,

Viceversa.Thatif

Unitsmust

ThisindicatesthattheMaclaurinseries(9-5-3)toandfunctionastheMaclaurinformula(9-5-2)oftheremainderterm(when).

Inthisway,wegetafunctionthepowerseriesexpansion:

.(9-5-4)

Itisthefunctionthepowerseriesexpression,if,thefunctionofthepowerseriesexpansionisunique.Infact,assumingthefunct