TITLE: If $\sum_{n=1}^\infty a_n$ is convergent, then $\sum_{n=1}^{\infty} a_n\sin(n)$
QUESTION [7 upvotes]: If the series $\sum_{n=1}^\infty a_n$ is convergent (absolutely convergent or conditionally convergent), then
  $$ \sum_{n=1}^\infty a_n \sin(n) $$
  is also convergent.

Any hint? today I'm not at my best. Thanks in advance.

REPLY [2 votes]: When the series is absouletely convergent, the proof is fairly straightforward as we have
$$
\sum_{n=1}^\infty |a_n| \geq \sum_{n=1}^\infty |a_n \sin(n)|
$$
as $ |\sin(n) | \leq 1 $ for all n. For conditionally convergent series, I'm not too sure on how it can be proven. Indeed, for a lot of conditionally convergent series, doing anything to the $a_n$ terms can cause the new series to not converge.