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Closed form for geometricish series (index squared in the exponent)?

  1. Dec 10, 2013 #1
    Closed form for "geometricish" series (index squared in the exponent)?

    Hi all,

    Is there a nice closed form for the following series?

    [itex]\sum_{k=0}^n x^{k^2}[/itex]

    Even a decently tight upper bound and lower bound would be nice (obviously it is bounded by the corresponding geometric series [itex]\sum x^k[/itex], but is there anything better?).

    Thanks!
     
  2. jcsd
  3. Dec 10, 2013 #2
    Hi !
    As far as I know, there is no referenced closed form for the finite sum x^k² , k=0 to n.
    The closed form for the infinite sum x^k² k=0 to infinity, with -1<x<1, is =(theta(0 , x)+1)/2 , involving the Jacobi theta function.
     
  4. Dec 29, 2013 #3
    Thanks for the reply! It turned out that the plain old geometric upper bound was sufficient for what I was doing before, but now I need a lower bound for the following sum for [itex]x \in (0,1)[/itex].

    [tex]
    \sum_{k=0}^n [x^k + x^{k(k+1)/2} - 1]
    [/tex]

    Clearly as n goes to infinity, this sum goes to negative infinity, but first it will increase with n and then decrease. I want a lower bound that captures that beginning increasing part. I don't really understand much about the Jacobi theta function so I'm not sure if it would be useful here too...

    Essentially what I need is a closed form lower bound so that I can solve an inequality for n.
     
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