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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.

hardmath
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Comic Book Guy
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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. In that case it doesn't make any sense(at least to me) to define it for $q$ being anything other than natural numbers. – hrkrshnn Jul 06 '14 at 15:59
  • As with your other Question about Abel's rule, I'm suggesting fixing the broken link with this replacement: Abel's multiplication rule for series. – hardmath Jan 19 '22 at 16:53
  • I went ahead and made the replacement. Please review that I've not unintentionally changed your meaning. – hardmath Feb 01 '22 at 17:41

1 Answers1

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This problem is modified from the problem in the paper titled "On the Multiplication and Involution of Semi-Convergent Series" by Florian Cajori. Here is the link.

https://www.jstor.org/stable/2369794?seq=1#metadata_info_tab_contents

See P204.

Jk_0fjnewuif
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  • When answering with a link to an external site, it is the guideline to quote the most relevant material to be found at the URL and/or to summarize that in your own words so Readers can judge whether to follow the link (and to help reconstructing the link if it becomes broken). See the FAQ on how to write a good answer under Provide context for links. – hardmath Jan 19 '22 at 16:59