Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.6 - Alternating Series and Conditional Convergence - Exercises 10.6 - Page 603: 10

Answer

Converges

Work Step by Step

Let us consider $u_n=\dfrac{1}{\ln n}$ and $u_n$ refers to the all positive values of $n$. The given function implies that $\ln n$ is an increasing function, but the sequence $u_n$ is decreasing. Then, $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{1}{\ln n}=0$ Hence, the series converges by the Alternating Series Test.
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