# Polynomial series

Consider the following polynomial series:

$S(x) = \sum_{i=1}^{\infty}(-1)^{i+1}x^{i^{2}}$

Between 0 and 1, this looks like a well-behaved function - is there any way to write this function in this interval without using a series?

Given $0 < S(x) < 1$, I need to solve the equation for $x$ (in the 0 to 1 interval), but an analytic solution would be much nicer than a numerical one...

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You can turn that series into a theta function with little effort: en.wikipedia.org/wiki/Theta_function –  Dan Piponi Apr 23 '10 at 21:42

$S(x) = -\frac12 \theta(\frac12, \frac{\log x}{\pi i }) - 1$, where $\theta$ is Jacobi's theta function. I don't know of any nice algebraic methods to take inverses of modular forms (even if you restrict to real $x$, i.e., $\tau$ purely imaginary), so I'm not sure if this is the sort of answer you want.
As other correspondents have pointed out, this is essentially a theta function. You ask if you can write it in any other way. You can replace the infinite series by an infinite product :-) One gets $$1-2S(x)=\prod_{n=1}^\infty(1-x^{2m-1})^2(1-x^{2m}).$$ The equivalence is a special case of the Jacobi triple product: http://en.wikipedia.org/wiki/Jacobi_triple_product . Using this represention makes numerical computation of the function more difficult :-)