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 10.1 - Sequences - Exercises 10.1 - Page 570: 89


converges to $0$

Work Step by Step

Here, $\int_1^1 \dfrac{1}{n} dx$ and $[\ln x]_1^n=\ln n -\ln 1=\ln n-0=\ln n$ Let $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{\ln n}{n}$ But $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{\ln n}{n}=\dfrac{\infty}{\infty}$ This shows that the limit has an Indeterminate form so, we will use L-Hospital's rule. This implies that $\lim\limits_{n \to \infty} \dfrac{1}{n}=0$ Thus, {$a_n$} is Convergent and converges to $0$.
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