# Three series convergence problem [closed]

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?

## closed as off-topic by Saad, Ben, user21820, user 170039, Greg MartinDec 18 '18 at 19:20

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• What did you try? – José Carlos Santos Dec 18 '18 at 12:56
• @JoséCarlosSantos I'm not sure if I can use something like comparision for the series that $-a_n<a_n \sin(a_n) < a_n$ but I tried only this method – avan1235 Dec 18 '18 at 12:57

Yes, it converges, by the comparaison test and because$$(\forall n\in\mathbb{N}):\bigl\lvert a_n\sin(a_n)\bigr\rvert\leqslant\lvert a_n\rvert=a_n.$$
• Note that, since $\sum a_n$ is convergent, $\lim_{n\to\infty}a_n=0$. Since $a_n>0$, this implies that $a_n\in(0,\frac{\pi}{2})$ for $n$ large enough, so $\sin(a_n)>0$ for $n$ large enough. Therefore, you don't need the absolute value. (Of course, it makes the proof longer...) – Taladris Dec 18 '18 at 15:09
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.