University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 9 - Practice Exercises - Page 552: 32



Work Step by Step

Consider $a_n=\dfrac{\ln n}{\ln (\ln n)}$ Now, $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln (\ln n)}$ Here, $\lim\limits_{n \to \infty} \dfrac{\ln n}{\ln (\ln n)}=\dfrac{\infty}{\infty}$ This shows an Inderminate form of a limit, so apply L'Hospital's rule such that $\lim\limits_{x \to l}\dfrac{a(x)}{b(x)}=\lim\limits_{x \to l}\dfrac{a'(x)}{b'(x)}$ Thus, $\lim\limits_{n \to \infty}\dfrac{n \ln n}{n}=\infty$ Thus, the series Diverges.
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