Let $Z = Y-1$ where $Y \sim {\rm Geo}(p)$.
Calculate $G(x) = E[x^Z]$, the probability generating function of $Z$.
What I have done so far is:
$$E[x^Z] = E[x^Yx^{-1}] = x^{-1}E[x^Y] = x^{-1}\sum^\infty_{k=0}p(1-p)^{k-1}x^{k} = \sum^\infty_{k=0}p(1-p)^{k-1}x^{k-1}$$
However I know this isn't right.
If I can get some hints on how to handle $Z = Y-1$ I'm sure I can solve it.
Thanks in advance.
Edit
The key says:
For $x < \frac{1}{1-p}$ we have $G(x) = E[x^Z]=\sum^\infty_{k=0}p(1-p)^k\cdot x^k = \frac{p}{1-(1-p)x}$
From what I did I would get
$$\frac{p}{x(1-p)} + \sum^\infty_{k=1}p(1-p)^{k-1}x^{k-1} =\frac{p}{x(1-p)} + \frac{p}{1-(1-p)x}$$