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 577: 9

Answer

The interval of convergence is $(-1,1)$.

Work Step by Step

We apply the ratio test $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} \frac{(n+1)x^{n+1} }{nx^n}=|x|\lim _{n \rightarrow \infty} \frac{n+1}{n}=|x| $$ Hence, the series $\Sigma_{n=0}^{\infty} nx^n$ converges if and only if $|x|\lt1$. That is, the interval of convergence is $(-1,1)$.
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