Closed forms for two times series similar to geometric series, but with additional power

Question

Does anyone know a close form solutions to any of the following time series? (approximate upper bounds might as well work).

$$ \sum_{k=1}^T \frac{1}{2^{k^2}} $$ or $$ \sum_{k=1}^T k \frac{1}{2^{k^2}} $$

Answer 1

The first sum converges really fast as $T \to \infty$. For $T=10$ there are $35$ correct decimal digits. As $T \to \infty$ the first sum is related to Jacobi theta functions, and there is a closed-form solution in term of $\vartheta_3$: $$ \sum_{k=1}^\infty \frac{1}{2^{k^2}} = \frac{1}{2}\vartheta_3\left(0,\tfrac12\right)-\frac{1}{2} \approx 0.56446841360593857933472927427247566\dots $$ The closed-form is trivial from the definition of $\vartheta_3$ function. This value is a good upper bound of the partial sums for any finite $T$.

I didn't find similar closed-form solution to the second sum. The numerical value of the sum as $T \to \infty$ is the following: $$ \sum_{k=1}^\infty \frac{k}{2^{k^2}} \approx 0.6309205592551858647783240039079433700921514299217879868\dots $$