Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 34


Yes, the series $\Sigma_{n=1}^{\infty}a_n^{-1}$ converges.

Work Step by Step

Let $b_n=a_n^{-1}$; then applying the ratio test, we have $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{b_{n+1}}{b_{n}}\right|=\lim _{n \rightarrow \infty} \left|\frac{a_{n+1}^{-1}}{a_{n}^{-1}}\right|\\ =\lim _{n \rightarrow \infty} \left|\frac{a_{n } }{a_{n+1}}\right|=\frac{1}{4}\lt1 $$ Hence, the series $\Sigma_{n=1}^{\infty}a_n^{-1}$ converges.
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.