132Lecture19.pdf - Lecture 19 Taylor series and Maclaurin...

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Lecture 19: Taylor series and Maclaurin series. Theorem: if f ( x ) has a power series at a , then f ( x ) = X n =0 c n ( x - a ) n , where c n = f ( n ) ( a ) n ! . Definition: The series X n =0 f ( n ) ( a ) n ! ( x - a ) n is called Taylor series of f at a . In particular, when a = 0, the series X n =0 f ( n ) (0) n ! x n is called Maclaurin series. Examples: 1. Find the Maclaurin series for f ( x ) = e x , and find its interval of convergence. Since all the derivatives of e x are e x , thus f ( n ) (0) = e 0 = 1, hence e x = X n =0 f ( n ) (0) n ! x n = X n =0 x n n ! . To find where this series converges, use the ratio test and compute: lim n →∞ a n +1 a n = lim n →∞ n ! | x | n +1 ( n + 1)! | x | n = lim n →∞ | x | n + 1 = 0 . Thus the interval of convergence is all real numbers, or ( -∞ , ). 2. Find the Taylor series for f ( x ) = e x at a = 2. Since all the derivatives of e x are e x , thus f ( n ) (2) = e 2 , hence e x = X n =0 f ( n ) (2) n ! ( x - 2) n = X n =0 e 2 n ! ( x - 2) n . 3. Find the Maclaurin series for f ( x ) = sin( x ), and find its interval of convergence.
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