Answer
Diverges
Work Step by Step
Consider $a_n= \sin (\dfrac{1}{n})$ and $b_n=\dfrac{1}{ n}$
Now, $\lim\limits_{n \to \infty}\dfrac{a_n}{b_n} =\lim\limits_{n \to \infty}\dfrac{ \sin (\dfrac{1}{n})}{\dfrac{1}{ n}}$
Thus, we have to apply L-Hospital's rule.
Then $ =\lim\limits_{n \to \infty} \dfrac{\cos (1/n)}{1}$
or, $ \cos (0)=1 \ne 0 \ne \infty $
Here, $\Sigma_{n=1}^\infty \dfrac{1}{n}$ is a divergent series due to the p-series test with $p \leq 1$
Thus, the series diverges by the limit comparison test.