常见函数的泰勒级数展开.pdf

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1、22TAYLORSERIESTaylorSeriesforFunctionsofOneVariablefaxa′′()(−)2f(n−−1)()(axa−)n−122.1.fx()=+fafaxa()′()(−+)+⋅⋅⋅++R2!()n−1!nwhereR,theremainderafternterms,isgivenbyeitherofthefollowingforms:n()nnfx()(ξ−a)22.2.Lagrange’sform:R=nn!()nn−1fxx()(ξξ−−)(a)22.3.Cauchy’sform:R=n()n−1!T

2、hevaluex,whichmaybedifferentinthetwoforms,liesbetweenaandx.Theresultholdsiff(x)hascontinuousderivativesofordernatleast.IflimR=0,theinfiniteseriesobtainediscalledtheTaylorseriesforf(x)aboutxa.Ifa0,theseriesnn→∞isoftencalledaMaclaurinseries.Theseseries,oftencalledpowerseries,

3、generallyconvergeforallvaluesofxinsomeintervalcalledtheintervalofconvergenceanddivergeforallxoutsidethisinterval.SomeseriescontaintheBernoullinumbersBandtheEulernumbersEdefinedinChapter23,pagesnn142143.BinomialSeriesnn()−1nn()−−12()nnnn−−12n2n−3322.4.()axan+=+ax+ax+ax+⋅⋅⋅2!3

4、3!=+ann⎛n⎞ax−−12⎛n⎞an23+⎛n⎞axn−3⎜⎝12⎟⎠+⎜⎝⎟⎠xx⎜⎝3⎟⎠+⋅⋅⋅Specialcasesare22222.5.()axaa+=++2xx3322322.6.()axaaxa+=+++33xx443223422.7.()axa+=++464axax++axx−123422.8.()11+xx=−+x−x+x−⋅⋅⋅1

5、ERIES139113i135ii−12/2322.11.()11+xx=−+xx−+⋅⋅⋅1

6、∞0⎝1⎠3⎝1⎠5⎝⎝+1⎠⎩⎪⎭⎪23⎛x−11⎞⎛x−11⎞⎛x−1⎞122.20.lnx=⎜x⎟+2⎜x⎟+3⎜x⎟+⋅⋅⋅⋅x2⎝⎠⎝⎠⎝⎠Ser

7、iesforTrigonometricFunctions357xxx22.21.sinxx=−+−+−∞<

8、

9、x<315315()2n!21xx352x222nBxn−1n22.24.cotx=−−−−−−0<<

10、

11、xπx345945()2n!xxx246561Ex2nπn22.25.secx=++1++++

12、

13、x<224720()2n!21xx73531

14、x22()21n−−1Bxx21n−n22.26.cscx=++++++0<<

15、

16、xπx636015120,()2n!3571xx1