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: 61


The series $\sum_{n=1}^{\infty} \frac{1}{(\ln n)^4}$ diverges.

Work Step by Step

Since $\ln n\lt n^{1/4}$, then $(\ln n)^4\lt n $ and hence $$\frac{1}{(\ln n)^4}\gt \frac{1}{n}.$$ Now, since the series $\sum_{n=1}^{\infty} \frac{1}{n} $ is a divergent p-series, then the series $\sum_{n=1}^{\infty} \frac{1}{(\ln n)^4}$ diverges.
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