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chap11sec10 - 11.10 Taylor and Maclaurin Series...

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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   f   is represented by a power series   0 ( ) ( ) n n n f x c x a = = -   for all   x   in an open interval containing   a   then  ( ) ( ) ! n n f a c n =   and   ( ) 2 3 0 ( ) ( ) '( )( ) ''( )( ) '''( )( ) ( ) ( ) ! 0! 1! 2! 3! n n n f a f a f a x a f a x a f a x a f x x a n = - - - = - = + + + + L 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 f. ( ) 2 3 0 (0) (0) '(0) ''(0) '''(0) ( ) ! 0! 1! 2! 3! n n n f f f x f x f x f x x n = = = + + + + L 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  ( ) ( ) ! n n f a c n =  and determine the interval of  convergence for the resulting power series 2 3 ''( ) '''( ) ( ) '( )( ) ( ) ( ) 2! 3!
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