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1、9CHAPTER:POWERSERIESContents0Introduction1Maclaurin’sTheorem2Standardexpansionsx0IntroductionAfewminutesplaywithascientificcalculatorwillshowthatformostvaluesofthevariablex,thevaluesofthefunctionssinx,cosx,ex,lnx,andsoon,arenotrationalnumbersorrecognizab
2、leirrationalnumbersexpressibleasroots.Iftheinternallogicofacalculatororcomputerislimitedtothebasicarithmeticoperationsofaddition,subtraction,multiplication,anddivision,howdoesitcalculatethevalueoftheseandothersuchfunctions?Powerseriesisawayofrepresenting
3、thesefunctionsbypolynomialserieswhichwillallowustofindtheirrespectiveapproximatevaluestoanydesireddegreeofaccuracy.Definition:Apowerseriesisaseriesoftheforma0+a1x+a2x2+a3x3+...+anxn+...to¥,wherethecoefficientsa0,a1,a2,a3,...,areconstants.IfSn=a0+a1x+a2x2
4、+a3x3+...+anxn,andifSn®Sasn®¥thenthepowerseriesissaidtobeconvergentandhaveasumS.x1Maclaurin’sTheoremUndercertainconditions,afunctionf(x)canbeexpressedasaninfiniteseriesofascendingpowersofx.Assumingthatsuchaseriesexists,wemaywritef(x)=a0+a1x+a2x2+a3x3+...
5、+arxr+...whereaiareconstants.Differentiatingwithrespecttox,wehave,f’(x)=a1+2a2x+3a3x2+4a4x3+...+rarxr-1+...f’’(x)=2a2+3.2a3x+4.3a4x2+....+r(r-1)arxr-2+...f’’’(x)=3.2a3+4.3.2a4x+....+r(r-1)(r-2)arxr-3+............fr(x)=r(r-1)(r-2)...3.2.ar+...Substituting
6、x=0,wefindthatf(0)=a0a0=f(0)f’(0)=a1a1=f’(0)f’’(0)=2a2a2=f’’’(0)=3.2a3a3=................fr(0)=r(r-1)(r-2)...3.2arar=Substitutingtheseexpressionsforaibackintotheoriginalf(x),wegetf(x)=f(0)+f’(0)x+x2+x3+...+xr+...ThisresultiscalledMaclaurin’sTheoremandthe
7、seriesobtainedisknownastheMaclaurin’sSeriesforf(x).ItispossibletofindaMaclaurin’sseriesforanyfunctionf(x)whosederivativesf’(0),f’’(0),f’’’(0),...canbedetermined.Theseriesmustconvergetothesumf(x)inordertobeuseful.Hence,formanyfunctions,Maclaurin’sTheoremh
8、oldsonlywithinarestrictedrangeofvaluesofx.9Example1.1UseMaclaurin’stheoremtoobtaintheexpansionforf(x)=(1+x)nwherenisarealnumber.[1+nx+x2+...for-1
