Closed Form for Infinite Sum

1. Sep 22, 2014

stevendaryl

Staff Emeritus
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:

$\sum_n x^{n^2}$

for $0 < x < 1$

2. Sep 23, 2014

mathman

I strongly doubt it.

3. Sep 23, 2014

Matterwave

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...

4. Sep 23, 2014

Staff: Mentor

WA does not even have a closed form for x=1/2. Does not look good.

5. Sep 25, 2014

Matterwave

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.

6. Sep 26, 2014

economicsnerd

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...