Let $G(t,x)$ be the heat kernel $$ G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}. $$

Here is one approximation to $G(t,x)$:

$$ G_\epsilon(t,x)=e^{-t/\epsilon} \sum_{k=1}^\infty \left(\frac{t}{\epsilon}\right)^k \frac{1}{k!} G(k\epsilon,x). $$

The question is: can one show that for some $a>0$ and $C>0$

$$ \int_{\mathbb{R}}\left|G(t,x)-G_\epsilon(t,x)\right| d x \le e^{-t/\epsilon}+C \left(\frac{\epsilon}{t}\right)^{1/3},\quad \text{for $0<\epsilon/t\le a$}? $$

Thanks a lot for any hints!

-----EDIT------

Thanks Professor Lucia for his nice solution. It turns out that it is not sufficient for what we actually need. Here is the revised question:

Fix $\epsilon>0$. Is there a constant $C>0$ and $0<\beta<1/2$, such that

$$ \int_{\mathbb{R}}\left|G(t,x)-G_\epsilon(t,x)\right| d x \le e^{-t/\epsilon}+C \left(\frac{\epsilon}{t}\right)^{\beta},\quad \text{for all $t>0$}? $$

The exponent $1/3$ is mysterious. But any order $\beta<1/2$ will be fine. Or probably this will never happen?