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 - Practice Exercises - Page 636: 30


Absolutely convergent.

Work Step by Step

Integral Test states that the series converges when the integral $\int_{k}^\infty f(x) dx$ converges. Consider the series $a_n=\int_{2}^\infty \dfrac{1}{x(\ln x)^2} dx$ Suppose $p=\ln x$ and $dp=\frac{dx}{x}$ Thus, $a_n=\int_{\ln 2}^\infty \dfrac{dp}{p^2} dx$ $a_n=-\dfrac{1}{p}|_{\ln 2}^\infty$ $a_n=-\dfrac{1}{\infty}+\dfrac{1}{\ln 2}$ $a_n=\dfrac{1}{\ln 2}$ Therefore, the given series is Absolutely convergent.
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