University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.3 - The Integral Test - Exercises - Page 504: 19



Work Step by Step

Consider $f(x)=\dfrac{\ln x}{x}$ Now, take the integral test to find the convergence and divergence for the sequence. We have $\int_2^\infty \dfrac{\ln x}{x}dx= \lim\limits_{a \to \infty} \int_2^a \dfrac{\ln x}{x}dx$ or, $ \lim\limits_{a \to \infty} [\dfrac{1}{2}(\ln^2 x)]_2^a=\lim\limits_{a \to \infty} [\dfrac{1}{2}(\ln^2 a-\ln^2 2)]= \infty$ Hence, the sequence $\Sigma_{n=2}^\infty \dfrac{\ln n}{n}$ is Divergent.
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