## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 9

Converges

#### Work Step by Step

Given $$\sum_{n=1}^{\infty} \frac{10^n}{2^{n^2} }$$ By using the Ratio Test, we get: \begin{align*} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty} \frac{10^{n+1}}{2^{n^2+2n+1}} \frac{2^{n^2}}{10^n} \\ &= \lim _{n \rightarrow \infty} \frac{10}{2^{2n+1}}\\ &= 0<1 \end{align*} Thus the series converges.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.