I have a summation of the form:
$$ S = \sum_{i=1}^{\infty} \sqrt{\frac{2}{\pi i}} \vartheta_4\left(0,\exp\left(\frac{-2H^2}{i}\right)\right), $$ where $\vartheta_4(z=0,q)$ is the fourth jacobi theta function of the form
$$ \vartheta_4(0,q) = 1 + 2\sum_{i=0}^\infty (-q)^{i^2}. $$
Empirical results converge very precisely, i.e.,
$$ S^{\prime}=\sum_{i=1}^{H^3} \sqrt{\frac{2}{\pi i}} \vartheta_4\left(0,\exp\left(\frac{-2H^2}{i}\right)\right) = 2H - 1.16594. $$
I tried the sum for the range $H \in [50,200]$ and it converges almost perfectly to this point, even before $H^3$ is reached.
I am not even sure how to prove the existence of the summation. Maclaurin Cauchy test seems the most promising one, and the Wolfram Alpha suggests that
$$ \int_1^\infty \sqrt{\frac{1}{x}} \vartheta_4\left(0,\exp\left(\frac{-2H^2}{x}\right)\right)\mathrm{d}x \approx 0.12, $$
for $H = 50$. However, this is not really a proof and I am not in a position to calculate the integral myself let alone do the symbolic integration with $H$ as a variable. Moreover, Wolfram Alpha states it cannot be represented in common functions, even if $H$ is substituted with an integer.
So, my question is how to calculate this particular limit? Calculating it would also prove the existence of it as well :)
Thanks in advance!