finding moment generating function I am having a bit of trouble finding the moment generation function for 
$f(x)=(\frac{1}{2})^{x+1}$ for $x=0,1,2,3,\ldots$
I know that $M(x)=\sum e^{tx}(\frac{1}{2})^{x+1}$ which I have rearranged to make $\frac{1}{2} \sum (\frac{1}{2}e^t)^x$ but I am not sure how to simplify this further.

Indeed, the moment generating function is defined as
$$E\left[e^{tX}\right]=\sum_{i=0}^{\infty}e^{ti}P(X=i)=\sum_{i=0}^{\infty}e^{ti}\frac1{2^{i+1}}=\frac 12+\frac12\sum_{i=1}^{\infty}\left(\frac {e^t}2\right)^i.$$
This is a geometrical series with $q=\frac {e^t}2.$ This series is convergent if $0\le q<1$: $t<\ln(2).$ The sum of the series is then
$$E\left[e^{tX}\right]=\frac12\left[1+\frac{e^t}{2-e^t}\right]=\frac{1}{2-e^t}.$$
