The question is really quite simple - I would like to simplify the following series:
$\sum_{z=1}^{\infty}\sum_{y=1}^{\infty}\left(\frac{1}{2}\right)^y(1-e^{-y})^2e^{-y(x+z-2)}$
My approach: It is clear to me that we are dealing with a geometric series so I initially tried substituting the relevant formula as n tends to infinity. I'm not sure if this approach is correct however as we are multiplying each of these terms. Furthermore, the double summation is adding some additional confusion.
It's been a while since I've had to deal with these types of series (I usually rely on Mathematica) so it's possible that I'm forgetting some of the tricks involved. Any help or guidance would be much appreciated.