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: 35


$$\sum_{n=0}^{\infty}3 ^nx^ {n}$$ $|x|\lt 1/3$

Work Step by Step

Given $$ f(x)=\frac{1}{1-3 x}$$ 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}{1-3 x}&= \sum_{n=0}^{\infty}(3 x)^{n}\\ &= \sum_{n=0}^{\infty}3 ^nx^ {n} \end{align*} Such that $ |3x|\lt 1$, or $|x|\lt 1/3$
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