Hi this is my first question so far so I hope I'm doing it the right way. I'm trying to prove some result regarding the speed of congvergence of the Riemann sum
$\Phi(R,\delta):=\sum_{k \in \mathbb{Z}}\exp(-(k+\delta)^2R^2)(k+\delta)^mR^{m+1}$ to the Gaussian integral
$X_m :=\int_{\mathbb{R}} e^{-t^{2}}t^mdt$.
I used Fourier transformation on $\Phi$ and I'm left with controlling
$\lim_{R\rightarrow0}|\phi(R,\delta)-X_m|\le\lim_{R\rightarrow0}2\,C(m)\sum_{k\ge1}e^{-\pi k^2 /R^2}\left(1+\left(\frac{k}{R}\right)^m\right)$.
My question is to what function is $R\mapsto \sum_{k\ge1}e^{-\pi k^2 /R^2}\left(1+\left(\frac{k}{R}\right)^m\right)$ equivalent?
My guess was: $\lim_{R\rightarrow0}\frac{\sum_{k\ge1}\exp\left({-\pi k^2 /R^2}\right)\left(1+\left(\frac{k}{R}\right)^m\right)}{\exp(-\pi k^2/ R^2)}=\lim_{R\rightarrow0} \left(1+(\frac{1}{R})+\sum_{k\ge2}e^{-\pi (k^2-1) /R^2}\left(1+\left(\frac{k}{R}\right)^m\right) \right)$
If my guess is right, how do I get the convergence to zero of the left over series? Edit: I guess my first proposal to use dominated convergence was wrong. I'm pretty sure my claim is right but I do not how to give a formal justification.
Many thanks for your help! T.T