Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - Review - Exercises - Page 825: 11



Work Step by Step

The Comparison Test states that the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ is convergent if $p\gt 1$ and divergent if $p\leq 1$. $a_{n}=\frac{n}{n^{3}+1}\lt \frac{n}{n^{3}}=\frac{1}{n^{2}}$ Since, $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$ is convergent, $\sum_{n=1}^{\infty}\frac{n}{n^{3}+1}$ is also convergent. Hence, the series $\sum_{n=1}^{\infty}\frac{n}{n^{3}+1}$ is convergent.
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