Answer
Converges
Work Step by Step
Given: $a_{n+1}=\dfrac{1+\sin n}{n}a_n$ and $a_1=2$
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(1+\sin n)}{n}a_n}{a_n}|$
$\implies \lim\limits_{n \to \infty}|\dfrac{1+\sin n}{n}|=\lim\limits_{n \to \infty}|(\dfrac{1}{n})+(\dfrac{\sin n}{n})|=0 \lt 1$
Thus, the series converges by the ratio test.