ch11s10 - 11.10 Taylor and Maclaurin Series In this section...

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11.10 Taylor and Maclaurin Series In this section we will study a general procedure for deriving the power series of a function that has derivatives of all orders. The following theorem gives the form that every (convergent) power series must take. The Form of a Convergent Power Series If is represented by a power series for all in a open interval containing then and f 0 () ( ) n n n fx c x a = =− x a ! n n fa c n = 23 0 ' ( ) ' ' ( ) ' ' ' ( ) ( ) !0 ! 1 !2 !3 ! n n n f a faxa f axa x a n = −− − = ++ + + L 2 3 0 (0) (0) '(0) ''(0) '''(0) ! 1 ! n n n ff f x f x f x x n = == + + + + L ! n n c n = ''( ) '''( ) ' ( ) ( ) ( ) 2! 3! f a xa +− + − + The series is called the Taylor Series of the function f at a (or centered at a ). If a=0, then the series is the Maclaurin Series for . A function that is equal to the sum of its Taylor Series is called analytic. Guidelines for Finding a Taylor Series (1) Differentiate f(x) several times and evaluate each derivative at x=a. Try to recognize a pattern in these numbers. (2) Use the sequence developed in the first step to form the Taylor coefficients and determine the interval of convergence for the resulting power series L (3) Within this interval of convergence, determine whether the series converges to f(x).
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ch11s10 - 11.10 Taylor and Maclaurin Series In this section...

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