## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 11 - Infinite Series - 11.4 Absolute and Conditional Convergence - Preliminary Questions - Page 562: 1

#### Answer

The series $\sum_{n=0}^{\infty} \frac{(-1)^{n-1}}{ n^{2/3}}$ converges conditionally.

#### Work Step by Step

The series series $\sum_{n=0}^{\infty} \frac{(-1)^{n-1}}{ n^{2/3}}$ converges conditionally. We have the absolute series $$\sum_{n=0}^{\infty} |\frac{(-1)^{n-1}}{ n^{2/3}}|=\sum_{n=0}^{\infty} \frac{1}{n^{2/3}}$$ which is a divergent p-series as $p=2/3\lt 1$. The series $\frac{1}{ n^{2/3}}$ 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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