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.

Proof:

$(\rightarrow)$ $\sum a\_n$ 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.