For specific functions and $a_n >0$, we can say that $\sum a_n$ converges $\iff$ $\sum f(a_n)$ converges. I want to show this specifically for the case that $f(x)=sin(x)$. This is what I have thus far (which I admit is essentially nothing). Any direction would be welcome.
$(\rightarrow)$ $\sum a_n$ converges. We know that:
(i) the partial sums of $a_n$ are bounded.
$(\leftarrow)$ $\sum f(a_n)$ converges. We know that:
This is not a duplicate of a previous question that I asked. Before, I asked for a proof of this concept (if it could be proved). This, I am asking for help with a direct application of the concept.