## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 11 - Infinite Series - Chapter Review Exercises - Page 591: 47

converges

#### Work Step by Step

Given $$\sum_{n=2}^{\infty} \frac{n}{\sqrt{n^{5}+5}}$$ Compare with $\sum\frac{1}{n^{3/2}}$, a convergent series ($p>1$): \begin{align*} \lim_{n\to\infty } \frac{a_n}{b_n}&=\lim_{n\to\infty } \frac{n^{5/2}}{\sqrt{n^{5}+5}}\\ &=\lim_{n\to\infty } \frac{n^{5/2} /n^{5/2}}{\sqrt{1+5/n^{5}}}\\ &=1 \end{align*} Hence, $\sum_{n=2}^{\infty} \frac{n}{\sqrt{n^{5}+5}}$ also converges.

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