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I encountered this sum in my math class and I don't know how to get a closed form. I've tried using the residue theorem and it hasn't gone anywhere, although I'm guessing that's how one would be found. Any ideas?

Carl Schildkraut
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    there is nothing to solve, there is not a simple form – jimjim Feb 25 '16 at 18:23
  • How do you know? What do you mean? – Carl Schildkraut Feb 25 '16 at 18:24
  • solve has a different meaning, What i know is not important, i can not prove what i said, so i really should have not said it. In my unproved opinion is too ugly to have a closed form, but math is not about opinions. – jimjim Feb 25 '16 at 18:29
  • @CarlSchildkraut Put another way, what makes you believe there is a "nice" closed-form? (for some suitable definition of "nice"). – Clement C. Feb 25 '16 at 18:30
  • Ok. I edited my question to make it clearer that I'm looking for a closed form and not an abstract "solution". I feel like it should have a closed form, but as you said math is not about opinions. – Carl Schildkraut Feb 25 '16 at 18:30
  • @ClementC. I'm not sure if it has a closed form, but I think since it's such a simple series it probably should. If you don't think it does, why not? – Carl Schildkraut Feb 25 '16 at 18:31
  • Oh, I just have a completely heuristic argument: most things I have seen either provably have no simple closed forms or, if they have, have some closed form out of my reach. But this is in no way a proof: simply, if you knew for sure it has one (e.g., it is from an assignment) then this would definitely close the debate. – Clement C. Feb 25 '16 at 18:32
  • @ClementC. You might be right. The assignment only asked for a proof of convergence (very simple) and not a closed form, but I thought it was an interesting series. – Carl Schildkraut Feb 25 '16 at 18:35
  • Derivative is uglier that original, and doesnt exist an antiderivative writable in simple functions. Little hope for this series to have a closed form. Maybe writing interior as a series and trying to apply (if possible) some theorem to swap summation, or a change of variable... anyway seems very ugly. – Masacroso Feb 25 '16 at 18:40
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    @CarlSchildkraut: Even the simpler series $~\displaystyle\sum_{n=1}^\infty\frac1{3^n+1}~$ does not possess a meaningful closed form, but can only be expressed in terms of the special q-polygamma function. – Lucian Feb 26 '16 at 09:53

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