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Drew75
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It's normal you'd arrive at the wrong answer in this case. The problem is that your index is wrong. There are two definitions for the pdf of a geometric distribution. The one you use, where $E(X)=\frac{1}{p}$ is defined from 1 to infinity. At zero it is not defined. So, the generating function needs to take this into account, as well.

$$\pi(s)=E(S^X)=\sum^\infty_{i=1}q^{i-1}ps^i$$ $$= ps\sum^\infty_{i=1}(qs)^{i-1}=ps\sum^\infty_{i=0}(qs)^i$$ $$=\frac{ps}{1-qs}$$

If you use the alternative definition, where $P(Y=y)=q^ip$, then the pdf is defined at zero. In this case the generating function converges to $\frac{p}{1-qs}$.

As for what $s$ represents, as far as I know it represents nothing. Generating functions are derived functions that hold information in their coefficients. They are sometimes left as an infinite sum, sometimes they have a closed form expression. Take a look at the wikipedia article, which give some examples of how they can be used. Here and here.wiki article probability generating functions and wiki article generating functions

Drew75
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