Answer
Divergent
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.
