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: 70


The series $\sum_{n=1}^{\infty}\frac{\sin(1/n)}{\sqrt n} $ converges.

Work Step by Step

Since $\sin(1/n)\lt 1/n$, then $$\frac{\sin(1/n)}{\sqrt n}\lt\frac{1}{ n^{3/2}}$$ Since the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{3/2} } $ converges (as $3/2\gt 1$), then by the comparison test, the series $\sum_{n=1}^{\infty}\frac{\sin(1/n)}{\sqrt n} $ converges.
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