Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.4 Absolute and Conditional Convergence - Exercises - Page 563: 3


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

Work Step by Step

We have the absolute series $$\sum_{n=1}^{\infty} |\frac{(-1)^{n-1}}{ n^{1/3}}|=\sum_{n=0}^{\infty} \frac{1}{n^{1/3}}$$ which is a divergent p-series as $p=1/3\lt 1$. The series $\frac{1}{ n^{1/3}}$ is positive, decreasing and tending toward zero. Thus, the original series converges by the Alternative Series Test. Therefore, overall the series converges conditionally.
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