Three series convergence problem

Question

Having that $\sum_{n=1}^{\infty}a_n$ is a convergent series and we know that $a_n > 0$ can we say that $\sum_{n=1}^{\infty}a_n \sin(a_n)$ also converges?

Answer 1

Yes, it converges, by the comparison test and because$$(\forall n\in\mathbb{N}):\bigl\lvert a_n\sin(a_n)\bigr\rvert\leqslant\lvert a_n\rvert=a_n.$$

Answer 2

Since $\sum a_n$ converges, for each $\varepsilon >0$, there exists $N \in \mathbb N^*$ s.t. whenever $\mathbb N^* \ni n > N, p \in \mathbb N^*$, $\sum_{n+1}^{n+p} a_k < \varepsilon$. Then $$ \left\vert \sum_{n+1}^{n+p} a_k \sin (a_k) \right\vert \leqslant \sum_{n+1}^{n+p} \vert a_k \sin (a_k ) \vert \leqslant \sum_{n+1}^{n+p} a_k < \varepsilon, $$ hence $\sum a_n \sin(a_n)$ converges by Cauchy convergence principle.