Calculate $\sum\limits_{n=0}^{\infty}{\int\limits_{\frac{1}{2}}^{\infty}(1-e^{-t})^{n}e^{-t^2}dt}$

$$\mbox{Calculate}\quad\sum_{n = 0}^{\infty}\int_{1/2}^{\infty}\left(1 - {\rm e}^{-t}\right)^{n}{\rm e}^{-t^{2}}{\rm d}t$$

• Basically I don't know where to start.
• I was thinking of using Tonelli Theorem but I have no idea how to calculate this sum, and neither do I know how to solve this integral.
• I also tried to expand$\left(1 - {\rm e}^{-t}\right)^{n}$ but that seems to leadnowhere.

This is a problem from Introductory Measure Theory course so I was expecting some of those methods to work. Any help would be greatly appreciated.