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: 8


The radius of convergence is $4$.

Work Step by Step

We apply the ratio test $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ x^{3n+4} /64^{n+1} }{ x^{3n+1} /64^{n} }|= |\frac{ x^{3 } }{ 64}| $$ Hence, the series $\Sigma_{n=0}^{\infty} x^{3n+1} /64^{n}$ converges if and only if $\rho= |\frac{ x^{3 } }{ 64}| \lt1$. That is, the interval of convergence is $(-4,4)$ and the radius of convergence is $4$.
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