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: 7

Answer

converges to $0$.

Work Step by Step

As we know that a sequence converges when $\lim\limits_{n \to \infty}a_n$ exists. Consider $a_n=\dfrac{\ln (n^2)}{n}$ Re-write the given sequence as:$a_n=\dfrac{2 \ln n}{n}$ Apply limits to both sides. $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\dfrac{2 \ln n}{n}$ $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\frac{\infty}{\infty}$ Since, we can see that the limit has the form of $\frac{\infty}{\infty}$, use L-Hospital's rule. $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\dfrac{\frac{2}{n}}{1}$ $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\dfrac{2}{n}$ $\lim\limits_{n \to \infty}a_n=\dfrac{2}{\infty}$ $\lim\limits_{n \to \infty}a_n=0$ Therefore, the sequence converges to $0$.
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