MATH231 Lecture Notes 9

# MATH231 Lecture Notes 9 - Taylor Series 8.7 Definition:...

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Unformatted text preview: Taylor Series 8.7 Definition: Recall that, given an infinitely differentiable function f ( x ) the Taylor series for f ( x ) about the point c is summationdisplay k =0 d k f dx k ( c ) ( x- c ) k k ! Example: Suppose f ( x ) = ln( x ). Then f ( x ) = 1 x f ( x ) =- 1 x 2 f ( x ) = 2 x 3 f ( x ) =- 3 2 x 4 f ( v )( x ) = 4 3 2 x 5 f k ( x ) = (- 1) k +1 ( k- 1)! x k The Taylor series for f ( x ) = ln( x ) about the point c = 2 is summationdisplay k =0 d k f dx k ( c ) ( x- c ) k k ! = ln(2) + summationdisplay k =1 (- 1) k +1 ( k- 1)! k ! ( x- 2) k 2 k It is easy to check that the radius of convergence of this series is 2. Example: Suppose f ( x ) = cos( x ). Then f ( x ) =- sin( x ) f ( x ) =- cos( x ) f ( x ) = sin( x ) f ( x ) = cos( x ) 1 The Taylor Series about the point c = 0 is given by The Taylor series about the point c = 1 is cos(1)- sin(1)( x- 1)- cos(1)( x- 1) 2 / 2!+sin(1)( x- 1) 3 / 3!+cos(1)( x- 1) 4 / 4!+ . . ....
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## This note was uploaded on 04/07/2008 for the course MATH 231 taught by Professor Bronski during the Spring '08 term at University of Illinois at Urbana–Champaign.

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MATH231 Lecture Notes 9 - Taylor Series 8.7 Definition:...

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