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

[lasindi]

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

Hi all,

Is there a nice closed form for the following series?

$\sum_{k=0}^n x^{k^2}$

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

Thanks!

 

[JJacquelin]

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.

 

[lasindi]

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 $x \in (0,1)$.

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

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.