Bounds for the Green function of heat equation

Question

It is well known that the Green Function of the 1-D Heat equation is given by the density of the normal distribution, i.e. $$G(t,x,y)=\frac{1}{\sqrt{4 \pi t}} e^{\frac{-(x-y)^2}{4t}}.$$ Now, I was told that by "elementary calculations" one can approximate it by the following quantity $$R_\epsilon (t,x,y) =e^{-t/\epsilon} \sum_{n=1}^\infty \frac{(t/\epsilon)^n}{n ! } G(n\epsilon,x,y)$$ in the sense that for some $a,b>0$ we have $$\int_\mathbb{R}|R_\epsilon (t,x,y)-G(t,x,y)|\, dy \leq e^{-t/\epsilon} + a(\epsilon/t)^{1/3}$$ for $0<\epsilon /t<b.$ It is not clear to me where the $1/3$ part comes from or how $t/\epsilon$ becomes $\epsilon /t$. Any ideas? If you are interested in the background of this questions: These aprroximations appear in noise regularization for SPDEs.