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Closed Form for Infinite Sum

  1. Sep 22, 2014 #1

    stevendaryl

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    This isn't quite a calculus question, but it didn't seem right for any of the other mathematics forums, either.

    Does anybody if there is a closed form for the following infinite series:

    [itex]\sum_n x^{n^2}[/itex]

    for [itex]0 < x < 1[/itex]
     
  2. jcsd
  3. Sep 23, 2014 #2

    mathman

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    I strongly doubt it.
     
  4. Sep 23, 2014 #3

    Matterwave

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    Wolfram alpha gives it in terms of an elliptic theta function...no idea what that is:

    http://www.wolframalpha.com/input/?i=1+x+x^4+x^9+...

    Looking at the wikipedia page, it looks like these functions are defined through infinite series such as the one seen in the OP...
     
  5. Sep 23, 2014 #4

    mfb

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    WA does not even have a closed form for x=1/2. Does not look good.
     
  6. Sep 25, 2014 #5

    Matterwave

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    I just realized, you have to include the ...'s at the end of that URL or else it doesn't work just by clicking on the link.
     
  7. Sep 26, 2014 #6
    If you could derive one, my best guess of how would be by some clever combinatorial argument, viewing it as a generating function. Just a guess...
     
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