## 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: 3

Converges

#### Work Step by Step

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

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