Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.6 Power Series - Exercises - Page 578: 39


$$\sum_{n=0}^{\infty}(-1) ^nx^ {2n}$$ $ |x|\lt1$

Work Step by Step

Given $$ f(x)=\frac{1}{1+ x^2}$$ Since $$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n},\ \ \ \ |x|\lt 1\tag{1}$$ By using (1), we get \begin{align*} \frac{1}{1+x^2}&=\frac{1}{1-(- x^2)}\\ &= \sum_{n=0}^{\infty}(- x^2)^{n}\\ &= \sum_{n=0}^{\infty}(-1) ^nx^ {2n} \end{align*} Such that $ |x|\lt1$
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