As part of a larger physical model I am currently searching for a solution to the following expression, a numerical solution is fine as I am ultimately really after the numerical result. $\alpha$ is a physical value and I need solutions for upto $n = 200000$
$ \sum\limits_{k=0}^{n}\frac{1}{2k+1} \lim_{\sigma \to 0} \frac{d^{2k}}{d\sigma^{2k}} \left(\exp(0.5[\alpha-\sigma]^2) . \mathrm{erfc}([\alpha-\sigma]/\sqrt2) \right), \alpha \in R $
$\mathrm{erfc}$ is the complementary error function.
Obviously an iterative solution would be best as it allows to find solutions for all $n$ in one run.