Calculus TAYLOR AND MACLAURIN SERIES Give a uctio ( ad a poit a, we wish to approimate ( i the eighborhood o a by a polyomial o degree. c c ( a c( a c( a P ( c ( a We have coeiciets to choose. We require that P ( match the uctio ( ad its irst derivatives at a. ( (a P (a, (a P (a, (a P (a, ( ( a P ( a (a P (a c ( c c( a c( a c( a at a (a P (a c ( c c ( a c ( a c ( a at a (a P (a c ( c c ( a 4c ( a ( c ( a at a 4 (a P (a c ( c 4c ( a ( ( c ( a at a ( ( ( a P ( a 4! c ( ( ( ( c We solve or the coeiciets c, c (a c (a (a! I we deie The polyomial ( the the ormula c ( (a ( a, c!! ( ( a holds or all.! c c ( a c( a c( a P ( c ( a ( ( ( a ( a ( a ( a ( a ( a( a ( a ( a ( a ( a is!!!! called the th order Taylor polyomial or ( about a. Note that is the order o the derivative ot the umber o terms. I the th order derivative o ( is zero the the th order Taylor polyomial, P (, is ot ecessarily o degree. For istace the irst two Taylor polyomials o cos about are P ( ad P (. I a the polyomial is called the MacLauri polyomial. As icreases we match higher order derivatives o ( with P ( at a. We might epect that the approimatio P ( o ( improves as icreases but that is ot always the case. We hope that lim ( (. The lim ( P P The Taylor Series o ( about a is is deied to be the Taylor Series o ( about a ( ( a ( a!. I a it is called a MacLauri Series.
The Taylor Series is a power series about a. Thereore there are oly three possibilities or the iterval o covergece o the series.. The series coverges absolutely or all. The radius o covergece is.. The series coverges absolutely oly or a. The radius o covergece is zero.. The series coverges absolutely i a iterval about a, ( a R, a R ad diverges outside the iterval. The edpoits eed to be tested separately. R is called the radius o covergece. R For ay give Taylor Series or ( we eed to determie the ollowig:. What is the iterval o covergece or? We use the ratio test.. Does the series coverge to (? Eample: ( e. Fid the rd order MacLauri polyomial, P (. This is a polyomial epressio o degree or e about which matches the value o e ad the value o the irst ad secod ad third derivatives o e at with the values o the polyomial at. ( ( P ( ( (!! ( e, ( e, ( e, ( e P (!! y P P P. P ep(.. - y P vs ep( aw ay rom P ep( Fid the MacLauri series o e ( ( ( (. The MacLauri series o e is.!!!! Does the series coverge? lim lim. The series coverges absolutely or all by (! the ratio test. The questio remais, does the Taylor series coverge to (? Eample: Fid the Taylor series or si about
The Taylor series or ( about is ( si ( cos ( si cos (4 ( si ( ( cos ( (! 4 6 7 The Taylor series or si about is!! 4!! 6! 7! 7. Veriy this series coverges absolutely or all.!! 7! (! ( Eample: Fid the MacLauri series o cos The Taylor series or ( about is ( cos ( si ( cos ( si (4 ( cos ( ( si (! 4 6 7 The Taylor series or cos about is!! 4!! 6! 7! 4 6. Veriy this series coverges absolutely or all.! 4! 6! (! ( Eample: Fid the Taylor series o si about The Taylor series or ( about is ( si ( cos ( si cos (4 4 ( si ( cos ( ( (! (
I order to recogize powers ad actorials or ormulas it is useul ot to multiply out coeiciets. 4 6 7 7 The Taylor series or si about is!! 4!! 6! 7! ( (! Note that we could have id the Taylor Series directly by substitutio. Eample: Fid the Taylor series or l about. The Taylor Series o ( about is ( l ( ( ( (! ( ( (4 ( ( 4 ( 4 ( 4 Keep goig util you recogize a patter ad ca geerate a ormula or the th derivative o with respect ( to evaluated at, ( (! The Taylor series or l about is (! (! (. This series coverges absolutely or, diverges at ad coverges coditioally at. Eample: Fid the Taylor series o ( about ( ( ( 6 6 ( 6 ( ( 6 4 6 6 6 The Taylor series o ( is o course!!!! I a power series o coverges to ( o some ope iterval about the the power series is the Taylor series. Quiz: Fid the Taylor series o l( about.
You will eed to memorize the ollowig Taylor series about (MacLauri series. The iterval o covergece is also give. e (,!!! 7 si!! 7! cos 4 6! 4! 6! 4 l( 4 4 (! (! (, (, (, ] (, For homewor problems see http://voyager.dvc.edu/~lmoth/calc/hw4series.pd