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: 49


The series $\Sigma_{n=1}^{\infty} \frac{1}{n+\sqrt n} $ diverges.

Work Step by Step

The series $\Sigma_{n=1}^{\infty} \frac{1}{n }$ is a divergent p-series with $p= 1$. Now, by using the limit comparison test, we have: $$\lim_{n\to \infty} \frac{(1/(n+\sqrt n))}{1/n}=\lim_{n\to \infty} \frac{n}{n+\sqrt n}=1\gt0.$$ Hence, the series $\Sigma_{n=1}^{\infty} \frac{1}{n+\sqrt n} $ diverges.
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