Convergence of series containing sin

Question

If we know that a series a(n) converge, then do the series a(n). sin(n) also converge?

I tried some series and proved that if a(n) converges absolutely then the series a(n). sin(n) also converge.

But is it true in case of any convergent series?

Answer 1

Try $a(n) = \sin(n)/n$. $\sum_{n=1}^\infty \frac{\sin(n)}{n} = \frac{\pi - 1}{2}$.