Answer
$\Sigma sin(a_{n})$ converges.
Work Step by Step
Yes it is true. Since, $\Sigma a_{n}$ converges, $\lim\limits_{n \to \infty}a_{n}=0$
But $sinx\lt x$, for all $x\gt 0$ , so $a_{n}\gt sin (a_{n}$ for all $n$ .
Thus, by the comparison test $\Sigma a_{n}$ converges, so does $\Sigma sin(a_{n})$ converges.
