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: 69


$\sum_{n=1}^{\infty}\sin(1/n) $ diverges.

Work Step by Step

By the limit comparison test, consider $b_n=\frac{1}{n}$. Now, we have $$L=\lim_{n\to \infty} \frac{\sin(1/n)}{1/n}=\lim_{1/n\to 0} \frac{\sin(1/n)}{1/n}=1.$$ Since $L\gt 1$ and the series $\sum_{n=1}^{\infty}\frac{1 }{n } $ is a divergent p-series, then $\sum_{n=1}^{\infty}\sin(1/n) $ diverges.
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