Show that $(\sum a_{n}^{3} \sin n)$ converges given $\sum{a_n}$ converges

As $\{a_n\}$ is a positive sequence such that $\sum a_n$ converges, we have $0 \le a_n \le 1$ for $n$ large enough, say $n \ge M$.

Then for $n \ge M$

$$0 \le \vert a_n^3 \sin n \vert \le a_n^3 \le a_n$$

Hence $\sum a_n^3 \sin n$ converges absolutely.

Also a series $\sum a_n$ can be convergent while the sequence $\{n a_n\}$ diverges. Consider for example $a_n$ equals to $0$ if $n$ is not a square and equal to $1/n$ otherwise.