I am trying to show that the $qth$ power of the series $a_{1}\sin \theta +a_{2}\sin 2\theta +\ldots +a_{n}\sin n\theta +...$ is convergent whenever $q(1-r)<1$, r being the greatest number satisfying the relation $a_{n}\leq n^{-r }$ for all values of $n$.
My immediate thoughts were to multiply the series and rearrange the terms as Abel's rule and then follow along in the footsteps of the brilliant solution posted by André Nicolas here, but i am having a hard time determining what general term in the $qth$ product series looks like. Any help would be much appreciated.