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: 50


The series $\Sigma_{n=1}^{\infty}\frac{\cos^2 n}{n^2}$ converges.

Work Step by Step

Since $0\lt \cos^2 n\lt1$, then $$0\lt \frac{\cos^2 n}{n^2}\lt \frac{1}{n^2}.$$ Now, since $\Sigma_{n=1}^{\infty}\frac{1}{n^2}$ converges, then by the comparison test, the series $\Sigma_{n=1}^{\infty}\frac{\cos^2 n}{n^2}$ converges.
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