Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.7 Strategy for Testing Series - 11.7 Exercises - Page 786: 22



Work Step by Step

$$\lim\limits_{k\to\infty}|a_k|=\lim\limits_{k\to\infty}\left|\frac{1}{2+\sin{k}}\right|=\lim\limits_{k\to\infty}\frac{1}{2+|\sin{k}|}$$ The $|\sin{k}|$ oscillates between $0$ and $1$; therefore, the series oscillates between $\frac{1}{2}$ and $\frac{1}{3}$. Because the limit varies between two answers, it does not exist, thus the series $\sum_{n=1}^{\infty}\frac{1}{2+\sin{k}}$ is divergent.
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