# 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 AndrĂ© 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. – Comic Book Guy 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. – fermesomme Jul 6 '14 at 15:59