What's the sum of $\sum_{k=1}^\infty \frac{2^{kx}}{e^{k^2}}$? I already asked a similar question on another post:
What's the sum of $\sum \limits_{k=1}^{\infty}\frac{t^{k}}{k^{k}}$?
There are no problems with establishing a convergence for this power series:
$$\sum_{k=1}^\infty \frac{2^{kx}}{e^{k^2}}$$
but I have problems in determining its sum. 

$$\sum_{k=1}^{\infty}\frac{2^{kx}}{e^{k^{2}}} = -\frac{1}{2} + \frac{1}{2} \prod_{m=1}^{\infty} \left( 1 - \frac{1}{e^{2m}} \right) \left( 1+ \frac{ 2^x }{e^{2m-1} } \right) \left( 1 + \frac{1}{2^x e^{2m-1} }\right ). $$
