TAYLOR AND MACLAURIN SERIES
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1 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 + L 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, K P (a (a (a (a P (a P c ( c + c ( a + c ( a + L c ( a at a P (a c ( c + c ( a + c ( a + L c ( a at a c ( c + c( a + 4 c4 ( a L ( c ( a at a (a P (a c ( c + L at a M P 4 c4 ( a ( ( c ( a! c ( ( ( ( Kc We solve or the coeiciets c, K c (a c (a (a! I we deie The polyomial ( the the ormula c (a, K c!! holds or all.! c + c ( a + c ( a + L c ( a P c ( a ( ( a + ( a( a + ( a + ( a + L ( a ( a is!!!! called the th 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 Taylor polyomial, P (, is ot ecessarily o degree. That is why it is commoly called the th order Taylor polyomial. 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! (. I a it is called a MacLauri Series. HdTaylorSeries.doc Pro. L. A. Moth Page o
2 The Taylor Series is a power series about iterval o covergece o the series. a. Thereore there are oly three possibilities or the. 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 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( e Fid the MacLauri series o ( ( L (. The MacLauri series o e is L.!!! +! 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 HdTaylorSeries.doc Pro. L. A. Moth Page o
3 The Taylor series or ( about is (! si cos si cos si ( cos The Taylor series or si about is L!! 4!! 6! 7! 7 + ( + + L. 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 cos ( si The Taylor series or cos about is L!! 4!! 6! 7! 4 6 ( + +L. 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 si ( cos ( ( (! HdTaylorSeries.doc Pro. L. A. Moth Page o
4 I order to recogize powers ad actorials or ormulas it is useul ot to multiply out coeiciets The Taylor series or si about is L!! 4!! 6! 7! + ( ( ( +! Eample: Fid the Taylor series or l about. The Taylor Series o ( about is l ( ( (! ( ( ( ( 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 ( 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. HdTaylorSeries.doc Pro. L. A. Moth Page 4 o
5 You will eed to memorize the ollowig Taylor series about (MacLauri series. The iterval o covergece is also give. e L (,!!! 7 + ( si + + L (,!! 7! ( +! 4 6 cos + + L! 4! 6! 4 l( L L + ta L 7 ( (! ( + (, (, ] ( (, ( + ( + [, ] For homewor problems see HdTaylorSeries.doc Pro. L. A. Moth Page o
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