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: 45


The series $\Sigma_{n=1}^{\infty} \frac{n}{2n+1}$ diverges.

Work Step by Step

We have $$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\frac{n}{2n+1}=\lim\limits_{n \to \infty}\frac{1}{2 +1/n}=1/2\ne 0$$ Hence the series $\Sigma_{n=1}^{\infty} \frac{n}{2n+1}$ diverges.
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