介绍林级数高数chap 9.pptx

9.8MaclaurinandTaylorSeries;

PowerSeries

MaclaurinandTaylorSeriesWedefined:thenthMaclaurinpolynomialforafunctionasthenthTaylorpolynomialforfaboutx=x0as

MaclaurinandTaylorSeriesItisnotabigjumptoextendthenotionsofMaclaurinandTaylorpolynomialstoseriesbynotstoppingthesummationindexatn.Thus,wehavethefollowingdefinition.

Example1FindtheMaclaurinseriesfor(a)(b)(c)(d)

Example1FindtheMaclaurinseriesfor(a)(b)(c)(d)(a)WetaketheMaclaurinpolynomialandextendit.

Example1FindtheMaclaurinseriesfor(a)(b)(c)(d)(b)WetaketheMaclaurinpolynomialandextendit.

Example1FindtheMaclaurinseriesfor(a)(b)(c)(d)(c)WetaketheMaclaurinpolynomialandextendit.

Example1FindtheMaclaurinseriesfor(a)(b)(c)(d)(d)WetaketheMaclaurinpolynomialandextendit.

Example2FindtheTaylorseriesfor1/xaboutx=1.

Example2FindtheTaylorseriesfor1/xaboutx=1.WefoundinthelastsectionthatthenthTaylorpolynomialfor1/xaboutx=1is

Example2FindtheTaylorseriesfor1/xaboutx=1.WefoundinthelastsectionthatthenthTaylorpolynomialfor1/xaboutx=1isThus,theTaylorseriesfor1/xaboutx=1is

Youmightwonder:Whywouldweeverwanttoexpressaknownfunctionasasumofinfinitelymanyterms?Integratingfunctionswithoutelementary

antiderivativesSolvingdifferentialequationsApproximatingfunctionsbypolynomials

Scientistsdothistosimplifytheexpressionstheydealwith.Computerscientistsdothistorepresentfunctionsoncalculatorsandcomputers.

PowerSeriesinxIfareconstantsandxisavariable,thenaseriesoftheform