Originally Posted by

**RedXIII** Sorry, my mistake. It is indeed non-negative integers.

That definitively makes a lot more sense now.

Just the clarify my understanding, in the last step, the summation equals 2 because:

$\displaystyle \sum_{k\ge 0}\left(\dfrac{1}{2}\right)^k =\dfrac{1}{1-\frac{1}{2}}=\dfrac{1}{\frac{1}{2}}=2$?

Then for part B, would this be the solution?

$\displaystyle m(t)=E(e^t^k)=\sum_{k=0}^\infty e^t^k\left(\dfrac{1}{2}\right)^k=e^t \sum_{k=0}^\infty \left(\dfrac{e}{2}\right)^k=e^t\left(\frac{1}{1-\dfrac{e}{2}}\right)$

Thanks!