Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.4 Absolute and Conditional Convergence - Exercises - Page 563: 10


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

Work Step by Step

Since $\cos n\leq 1$ then we have $\frac{\cos n}{2^n}\leq \frac{1}{2^n}$. Now, the series $\Sigma_{n=1}^{\infty}\frac{1}{2^n}$ is a geometric series with $r=1/2\lt1$, which converges. Hence, the positive series $\Sigma_{n=1}^{\infty}|\frac{\cos n}{2^n}|$ converges. Thus, the series $\Sigma_{n=1}^{\infty}\frac{\cos n}{2^n}$ converges absolutley.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.