Another Question based on Abel's theorem of multiplication of series

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.

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What do you mean by $q$-th power of a series? –  Davide Giraudo Jul 21 '12 at 19:57
@DavideGiraudo I believe the qth power refers to multiplying the series by itself q number of times and obtaining a new qth power series of the original series. This is used as the scene for Abels' rule. In the question there is a link to the theorem, which may provide some more context. I hope this is helpfull. –  Hardy Jul 21 '12 at 21:16
By $q$th power of series, I believe that you are referring to [Cauchy product](en.wikipedia.org/wiki/Cauchy_product). In that case it doesn't make any sense(at least to me) to define it for $q$ being anything other than natural numbers. –  Fermé somme Jul 6 '14 at 15:59