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

  • Thread starter lasindi
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  • #1
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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!
 

Answers and Replies

  • #2
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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.
 
  • #3
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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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