Explicit form of this series expansion? I am considering the following series expansion:
$$f(k):=\sum_{n\geq 1} e^{-k n^2}$$
with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at least an upperbound of the type $|f(k)|\leq \frac{C}{k^p}$ for some constant $C$ and power $p$?
Thanks in advance!

A slight refinement:
$$ 0 < f(k) - e^{-k} = \sum_{k=2}^{\infty} e^{-kn^2} < \sum_{k=2}^{\infty} e^{-kn} = \frac{e^{-2k}}{1-e^{-k}}, $$
which is bounded by, say, $2e^{-2k}$ for $1-e^{-k}>1/2$, i.e. $k>\log{2}$. Hence, in the Poincaré-type asymptotics,
$$ f(k) = e^{-k} + O(e^{-2k}). $$

Aside: The $(\pi/k)^{1/2}$ bound is useful when $k$ is positive and close to zero: it can be understood in terms of the Poisson summation formula:
$$ \sum_{k = -\infty}^{\infty} e^{-\pi k n^2} = \frac{1}{\sqrt{k}} \sum_{k=-\infty}^{\infty} e^{-\pi n^2/k} $$
