Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - Chapter Review Exercises - Page 592: 56


converges conditionally

Work Step by Step

Given $$\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt[3]{n}+2 n}$$ Since $$\sum_{n=1}^{\infty}\left|a_{n}\right|=\sum_{n=1}^{\infty} \frac{1}{n^{1 / 3}+2 n}$$ Using the comparison test, we compare with $ \sum \frac{1}{n} $, a divergent series: \begin{align*} \lim\limits_{n \to \infty}\frac{n}{n^{1 / 3}+2 n}=1 \end{align*} Hence $\sum_{n=1}^{\infty}\left|a_{n}\right| $ diverges. On the other hand, since $ a_n =\dfrac{1}{\sqrt[3]{n}+2 n} $ is decreasing and $\lim_{n\to \infty } a_n=0$, then $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt[3]{n}+2 n}$ converges. Hence, $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{1 / 3}+2 n}$ converges conditionally.
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