Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.10 Taylor and Maclaurin Series - 11.10 Exercises - Page 812: 51


(a) $\frac{1}{\sqrt {1-x^{2}}}=1+\Sigma_{n=1}^{\infty}\frac{^{2n}}{2^{n}n!}$ (b) $sin^{-1}x=x+\Sigma_{n=1}^{\infty}\frac{^{2n+1}}{(2n+1)2^{n}n!}$

Work Step by Step

(a) $\frac{1}{\sqrt {1-x^{2}}}=(1-x^{2})^{-1/2}=(1+(-x^{2}))^{-1/2}$ $(1+(-x^{2}))^{-1/2}=\Sigma_{n=1}^{\infty}(-1/2)n(-1)^{n}x^{2n}$ $\frac{1}{\sqrt {1-x^{2}}}=1+\Sigma_{n=1}^{\infty}\frac{^{2n}}{2^{n}n!}$ (b) From part (a), we have $\frac{1}{\sqrt {1-x^{2}}}=1+\Sigma_{n=1}^{\infty}\frac{^{2n}}{2^{n}n!}$ The derivative of $sin^{-1}x$ is $\frac{1}{\sqrt {1-x^{2}}}$ $\int (1+\Sigma_{n=1}^{\infty}\frac{^{2n}}{2^{n}n!})dx=c+x+\Sigma_{n=1}^{\infty}\frac{^{2n+1}}{(2n+1)2^{n}n!}$ Set $x=0$ to find $c$ $sin^{-1}(0)=c+0+0$ $c=0$ Thus, $sin^{-1}x=x+\Sigma_{n=1}^{\infty}\frac{^{2n+1}}{(2n+1)2^{n}n!}$
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