Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - Review - Exercises - Page 804: 46

Answer

$cosx=\frac{1}{2}\Sigma_{n=0}^{\infty}(-1)^{n}[\frac{1}{(2n)!} (x-\frac{\pi}{3})^{2n}-\frac{\sqrt 3}{(2n+1)!}(x-\frac{\pi}{3})^{(2n+1)}]$

Work Step by Step

Taylor series: $f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+...$ $cosx=\frac{1}{2}-\frac{\sqrt 3}{2} (x-\frac{\pi}{3})-\frac{1}{2.2!}(x-\frac{\pi}{3})^{2}-...$ $cosx=\Sigma_{n=0}^{\infty}(-1)^{n}[\frac{1}{2(2n)!} (x-\frac{\pi}{3})^{2n}-\frac{\sqrt 3}{2.(2n+1)!}(x-\frac{\pi}{3})^{(2n+1)}]$ Hence, $cosx=\frac{1}{2}\Sigma_{n=0}^{\infty}(-1)^{n}[\frac{1}{(2n)!} (x-\frac{\pi}{3})^{2n}-\frac{\sqrt 3}{(2n+1)!}(x-\frac{\pi}{3})^{(2n+1)}]$
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