Consider the following polynomial series:

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

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...

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...