If $\{a_n\}$ is a sequence of positive terms such that the series $$\sum_{n=1}^\infty a_n$$
coverges, does the series $$\sum_{n=1}^\infty \sin a_n$$ also converge?
I believe that limit comparison test is necessary but I'm not sure how to use it here
Best Answer
Since $\lim\limits_{k \to \infty} a_k=0$, $$\tag 1\lim_{k \to \infty} \frac{\sin a_k}{a_k}=1$$
Thus $\sum_k |\sin a_k|$ converges $\iff \sum_k |a_k|=\sum_k a_k$ does. By your hypothesis and the above, $\sum_k \sin a_k$ will be absolutely convergent, so it will converge.