Answer
Divergent
Work Step by Step
To use the Direct Comparison Test, we need a known series $\Sigma_{n=1}^{\infty}b_{n}$ for the purpose of comparison.
We will choose $b_{n}=\frac{1}{n}$
The Comparison Test states that the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ is convergent if $p\gt 1$ and divergent if $p\leq 1$.
Now,$a_{n}=sin(\frac{1}{n})$ and $b_{n}=\frac{1}{n}$
$\lim\limits_{n \to \infty}\frac{a_{n}}{b_{n}}=\lim\limits_{n \to \infty}\frac{sin(\frac{1}{n})}{\frac{1}{n}}$
Let $t=\frac{1}{n}$, also note that $t \to 0$ as $n \to \infty$
$\lim\limits_{n \to \infty}\frac{a_{n}}{b_{n}}=\lim\limits_{t \to 0}\frac{sint}{t}$
Since, the limit is of the form $\frac{0}{0}$ , we can apply L Hospital's Rule
$=\lim\limits_{t \to 0}\frac{cost}{1}$
$=cos (0) $
$=1\ne 0 \ne \infty$
The given series diverges by limit comparison test because $\Sigma _{n=1}^{\infty} \frac{1}{n}$ is diverging.
