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: 51



Work Step by Step

We have the given series $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{4 n+9}$ We apply the limit comparison test with $b_n=(\frac{1}{n^{1/2}})$ (a divergent p-series with $p=1/2\lt 1$): $L=\lim_{n\rightarrow\infty} \frac{a_n}{b_n}=\lim_{n\rightarrow\infty}\frac{\sqrt{n}}{4 n+9}\times n^{1/2}=\lim_{n\rightarrow\infty}\frac{n}{4n+9}=1/4$ Since $L=1/4\gt 0$, our starting series diverges.
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.