Answer
$\sum_{n=1}^{\infty}\sin(1/n) $ diverges.
Work Step by Step
By the limit comparison test, consider $b_n=\frac{1}{n}$. Now, we have
$$L=\lim_{n\to \infty} \frac{\sin(1/n)}{1/n}=\lim_{1/n\to 0} \frac{\sin(1/n)}{1/n}=1.$$
Since $L\gt 1$ and the series $\sum_{n=1}^{\infty}\frac{1 }{n } $ is a divergent p-series, then $\sum_{n=1}^{\infty}\sin(1/n) $ diverges.
