Answer
$\Sigma a_{n}b_{n}$ converges.
Work Step by Step
Yes, it is true. Since $\Sigma a_{n}$ and $\Sigma b_{n}$ converges, $\lim\limits_{n \to \infty}a_{n}=0$ and $\lim\limits_{n \to \infty}b_{n}=0$
By the laws of arithmetic, since all the terms are positive, $a_{n}b_{n}\lt a_{n}$ (or $b_{n}$ ), so $a_{n}\gt sin (a_{n})$ for all $n$ .
Thus, by the comparison test $\Sigma a_{n}$ converges and so $\Sigma a_{n}b_{n}$ converges.
