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. 

There is this Jacobi theta function:
$$
\vartheta_3 \biggl(\frac{i}{2} x \operatorname{ln} (2),\operatorname{e} ^{-1}\biggr) = \sum_{k = -\infty}^{\infty} \operatorname{e} ^{-k^{2}} 2^{k x}
$$
But you stopped half-way through, so yours is not such a common one.  Yours is a "partial theta function"
