Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - Chapter Review Exercises - Page 592: 65


The series $\Sigma_{n=1}^{\infty} \frac{n^5}{5^n}$ converges.

Work Step by Step

To apply the ratio test, we have: $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} \frac{(n+1)^5/ (5)5^n}{n^5/5^n}=\frac{1}{5}\lim _{n \rightarrow \infty} \frac{(n+1)^5}{n^5}=\frac{1}{5}\lt 1 $$ Hence, the series $\Sigma_{n=1}^{\infty} \frac{n^5}{5^n}$ converges.
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