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: 9


converges to $1$.

Work Step by Step

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