## Cannot tell when a probability generating function converges for |s|<1

Hi, I have a problem that is already solved... I thought 3 of the 4 functions were probability generating functions, but I got one wrong and don't know why.

The solution says $g(s)=1+s-s^2$ is not a probability generating function. However, g(1)=1 and I think g(s) converges to 1 for |s|<1. Isn't that correct? If so, what is it that invalidates this function as a probability generating function?

The solution says that $g(s)=(1/3)*(1+s+s^4)$ and $g(s)=(2-s^2)^{-1}$ are prob. gen. functions and that $g(s)=1+s-s^2$ and $g(s)=(1/2)(1+s+s^3)$ are not (I know the last one is not because g(1)=3/2).

Thanks a lot for your help! :) And if you could also give me an explanation of other things I need to look for in a function to tell if it is a p.g.f., I would really appreciate that.
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 If $1 + s - s^2$ were a probability generating function, then we would have $P(2) = -1$.