Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.2 Summing and Infinite Series - Exercises - Page 547: 19



Work Step by Step

Given $$ \frac{0}{1}-\frac{1}{2}+\frac{2}{3}-\frac{3}{4}+\cdots= \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(n-1)}{n}$$ Since \begin{align*} \lim _{n \rightarrow \infty} \frac{(-1)^{n-1}(n-1)}{n}&=\lim _{n \rightarrow \infty} (-1)^{n-1}(1-1/n) \\ &=\begin{cases} 1&n\ \text{even}\\ -1&n\ \text{odd} \end{cases} \end{align*} Since the $n$th term $a_{n}$ does not converge to zero, thus the series diverges.
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