University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.4 - Comparison Tests - Exercises - Page 509: 46



Work Step by Step

Consider $a_n= \tan (\dfrac{1}{n})$ and $b_n=\dfrac{1}{ n}$ Now, $\lim\limits_{n \to \infty}\dfrac{a_n}{b_n} =\lim\limits_{n \to \infty}\dfrac{ \tan (\dfrac{1}{n})}{\dfrac{1}{ n}}$ Thus, we have to apply L-Hospital's rule. Then $ =\lim\limits_{n \to \infty} \dfrac{\sec^2 (1/n)}{1}$ or, $ \sec (0)=1 \ne 0 \ne \infty $ Here, $\Sigma_{n=1}^\infty \dfrac{1}{n}$ is a divergent series due to the p-series test with $p \leq 1$ Thus the series diverges by the limit comparison test.
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.