## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 557: 53

#### Answer

The series converges.

#### Work Step by Step

We have the given series $\sum_{n=1}^{\infty} \frac{n^{2}-n}{n^{5}+n}=\sum_{n=1}^{\infty} \frac{n-1}{n^{4}+1}$ We apply the limit comparison test with $b_n=(\frac{1}{n^3})$ (a convergent p-series with $p=3\gt 1$): $L=\lim_{n\rightarrow\infty} \frac{a_n}{b_n}=\lim_{n\rightarrow\infty}\frac{n-1}{n^{4}+1}\times(n^3)=\lim_{n\rightarrow\infty}\frac{n^4-n^3}{n^{4}+1}=\lim_{n\rightarrow\infty}\frac{1-1/n}{1+1/n^4}=1$ Since $L=1$, our starting series converges.

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