This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 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 ( ) ( ) 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! 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! 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 ( ) ( ) !...
View Full Document