I need to deal some of question of this type recent,but I always get stuck.

ex.

$\displaystyle \sum_{m=k}^{(tk^2)/2}\frac{1}{\sqrt{(\pi\frac{tk^2}{2})}}*exp{\frac{-m^2}{2tk^2}}$

tons of thanks in advance,if anyone has ideas

my teacher said it approximately equals to $\displaystyle \sqrt{\frac{2}{\pi}}\int_{\frac{1}{\sqrt{t}}}^{inf inity} e^{-x^{2}}\,dx$

but I have no idea how to convert it.