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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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!+ . . ....
View Full Document

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.

Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online