Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 557: 68


The series $\sum_{n=1}^{\infty}\frac{2+(-1)^n}{n^{3/2} } $ converges.

Work Step by Step

We have the series: $\sum_{n=1}^{\infty}\frac{2+(-1)^n}{n^{3/2} } $ Note that when $n$ is odd, then the series becomes $\frac{2-1}{n^{3/2}}=\frac{1}{n^{3/2}}$, a convergent p-series with $p=3/2\gt 1$. If $n$ is even, then we get $\frac{2+1}{n^{3/2}}=\frac{3}{n^{3/2}}$, which is also convergent. Thus, our series converges as well.
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