Answer
The series $\sum_{n=1}^{\infty}\frac{\sin(1/n)}{\sqrt n} $ converges.
Work Step by Step
Since $\sin(1/n)\lt 1/n$, then $$\frac{\sin(1/n)}{\sqrt n}\lt\frac{1}{ n^{3/2}}$$
Since the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{3/2} } $ converges (as $3/2\gt 1$), then by the comparison test, the series $\sum_{n=1}^{\infty}\frac{\sin(1/n)}{\sqrt n} $ converges.
