常见函数的泰勒级数展开

22TAYLORSERIESTaylorSeriesforFunctionsofOneVariable22.1.fxfafaxafaxafn()()()()()()!(=+′−+′′−+⋅⋅⋅+22−−−−+−111)()()()!axanRnnwhereRn,theremainderafternterms,isgivenbyeitherofthefollowingforms:22.2.Lagrange’sform:Rfxannnn=−()()()!ξ22.3.Cauchy’sform:Rfxxannnn=−−−−()()()()()!ξξ11Thevaluex,whichmaybedifferentinthetwoforms,liesbetweenaandx.Theresultholdsiff(x)hascontinuousderivativesofordernatleast.Iflim,nnR→∞=0theinfiniteseriesobtainediscalledtheTaylorseriesforf(x)aboutxa.Ifa0,theseriesisoftencalledaMaclaurinseries.Theseseries,oftencalledpowerseries,generallyconvergeforallvaluesofxinsomeintervalcalledtheintervalofconvergenceanddivergeforallxoutsidethisinterval.SomeseriescontaintheBernoullinumbersBnandtheEulernumbersEndefinedinChapter23,pages142143.BinomialSeries22.4.()()!()()axanaxnnaxnnnnnnn+=++−+−−−−122121233123312!axanaxnannnn−−−+⋅⋅⋅=+⎛⎝⎜⎞⎠⎟+⎛⎝⎜⎞⎠⎟xxnaxn2333+⎛⎝⎜⎞⎠⎟+⋅⋅⋅−Specialcasesare22.5.()axaaxx+=++222222.6.()axaaxaxx+=+++332233322.7.()axaaxaxaxx+=++++443223446422.8.()111234+=−+−+−⋅⋅⋅−xxxxx1x122.9.()1123452234+=−+−+−⋅⋅⋅−xxxxx1x122.10.()113610153234+=−+−+−⋅⋅⋅−xxxxx1x113813922.11.()111213241352461223+=−+−+⋅⋅⋅−xxxx/iiiiii1x122.12.()1112124132461223+=+−+−⋅⋅⋅xxxx/iiii1x122.13.()111314361473691323+=−+−+⋅⋅⋅−xxxx/iiiiii1x122.14.()1113236253691323+=+−+−⋅⋅⋅xxxx/iiii1x1SeriesforExponentialandLogarithmicFunctions22.15.exxxx=++++⋅⋅⋅12323!!∞x∞22.16.aexaxaxaxxa==++++⋅⋅⋅lnln(ln)!(ln)!12323∞x∞22.17.ln()1234234+=−+−+⋅⋅⋅xxxxx1x122.18.1211357357ln+−⎛⎝⎜⎞⎠⎟=++++⋅⋅⋅xxxxxx1x122.19.lnxxxxxxx=−+⎛⎝⎜⎞⎠⎟+−+⎛⎝⎜⎞⎠⎟+−+⎛211131115113⎝⎝⎜⎞⎠⎟+⋅⋅⋅⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪5x022.20.lnxxxxxxx=−⎛⎝⎜⎞⎠⎟+−⎛⎝⎜⎞⎠⎟+−⎛⎝⎜⎞⎠⎟[phone]⋅⋅⋅⋅x12SeriesforTrigonometricFunctions22.21.sin!!!xxxxxx=−+−+−∞∞35735722.22.cos!!!xxxxx=−+−+−∞∞124624622.23.tan()xxxxxBxnnnn=+++++−−357222321517315221112()!n+||xπ222.24.cot()!xxxxxBxnnnn=−−−−−−−1345294522352210||xπ22.25.sec()!xxxxExnnn=++++++125246172022462||xπ222.26.csc,()xxxxxBnn=+++++−−16736031151202213521xxnn212−+()!0||xπ22.27.sin−=++++1357123132451352467xxxxxiiiiii||x122.28.cossin−−=−=−+++⎛⎝⎜⎞11352212313245xxxxxππii⎠⎠⎟||x1TAYLORSERIES14022.29.tan||−=−+−+±−+−1357353571211315xxxxxxxxxπ+++−−⎧⎨⎪⎩⎪(,)ififxx1122.30.cottan||−−=−=−−+−⎛⎝⎜⎞⎠⎟113522351xxxxxxpππππ+−+−==−⎧⎨11315011135xxxpxpx(,)ifif⎪⎪⎪⎩⎪⎪22.31.seccos(/)−−==−++⋅[phone]xxxxxπiii⎛⎛⎝⎜⎞⎠⎟||x122.32.cscsin(/)−−==++⋅[phone]xxxxxiii||x1SeriesforHyperbolicFunctions22.33.sinh!!!xxxxxx=++++−∞∞35735722.34.cosh!!!xxxxx=++++−∞∞124624622.35.tanh()(xxxxxnnn=−+−+−−−3571223215173151221))()!Bxnnn212−+||xπ222.36.coth()(xxxxxBxnnnn=+−++−−−134529451235122122n)!+0||xπ22.37.sechxxxxExnnnn=−+−+−1252461720122462()()!++||xπ222.38.cschxxxxxnn=−+−+−−1673603115120122352,()(112112−+−)()!Bxnnn0||xπ22.39.sinh−=−+−+135723132451352467xxxxxiiiiiiiii|||ln||xxxx±+−+121221324413524624iiiiiiiii66116xxx−⎛⎝⎜⎞⎠⎟+−−⎡⎣⎢⎤⎦⎥⎧⎨⎪⎪⎩⎪⎪ifif22.40.coshln()−=±−++12421221324413524xxxxiiiiiiii6660161ixxx+⎛⎝⎜⎞⎠⎟⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪+−−ifcosh,ifcosh,−⎡⎣⎢⎤⎦⎥101xx22.41.tanh−=++++1357357xxxxx|x|122.42.coth−=++++13571131517xxxxx|x|1MiscellaneousSeries22.43.exxxxxsin=++−−[phone]−∞∞x22.44.eexxxxcos=−+−+⎛⎝⎜⎞⎠⎟12631720246−∞∞xTAYLORSERIES14122.45.exxxxxtan=+++++12238234||xπ222.46.exxxxxxnxxnnsinsin(/)/=++−−++235623309024πnn!+−∞∞x22.47.exxxxnxnxnncoscos(/)!/=+−−+++13624342π−∞∞x22.48.ln|sin|ln||xxxxxBxnn=−−−−−−246212618028352nnnn()!2+0||xπ22.49.ln|cos|(xxxxxn=−−−−−−−24682122124517252022nnnnBxnn−+122)()!||xπ222.50.ln|tan|ln||(xxxxxnn=+++++24622379062283522−−−+1212)()!Bxnnnn02||xπ22.51.ln()()()111112212133++=−++++−xxxxx||x1ReversionofPowerSeriesSuppose22.52.yCxCxCxCxCxCx=++++++12233445566then22.53.xCyCyCyCyCyCy=++++++12233445566where22.54.cC111=22.55.cCc1322=−22.56.cCccc15322132=−22.57.cCcccccc1741232312455=−−22.58.cCccccccccccc1951224123213524122631421=+−+−3322.59.cCcccccccccccc1116132512331334122784728=++−33214612224252842−−−ccccccTaylorSeriesforFunctionsofTwoVariables22.60.fxyfabxafabybfabxy(,)(,)()(,)()(,)!=+−+−+12{{()(,)()()(,)()xafabxaybfabybxxxy−+−−+−222ffabyy(,)}+wherefabfabxy(,),(,),…denotepartialderivativeswithrespecttox,y,…evaluatedatxa,yb.TAYLORSERIES