Can probability generating function diverge in 1?

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Let X be a discrete random variable taking value in Z>=0 and G(t)=E(t^X) be it's PHF.

It's true that left limit of G in 1 is equal to 1 , i.e.

lim G(t)=1 (t -> 1 from the left)

But is there an example in which the value of G(1) is not defined ?