I'm calcuting the moment generating function for the discrete random variable $X=\mathbb{N}$, assuming probability $p(n)=\frac{e^{-\frac{1}{2}}}{2^n n!}$. Then I get $$ g(t)=e^{-\frac{1}{2}} \sum\limits_{n=0}^\infty \frac{e^{tn}}{2^n n!} \,\,\, t \in \mathbb{C} $$
Now, is there a simple expression for the sum of the above series?
Yes, indeed there is. This is $$e^{-\frac 12}\sum_{n=0}^{\infty} \frac{(e^t/2)^n}{n!} \\ = e^{-\frac 12} e^{e^t/2} \\ =e^{\frac{e^t-1}{2}}$$