Solution
= Solution
By the definition of a <probability generating function> and the power series for the <exponential function>,
$$
G_X(s)=\mathbb E[s^X]
=e^{-\lambda}\sum_{r=0}^{\infty}\frac{(\lambda s)^r}{r!}
=\boxed{e^{\lambda(s-1)}}.
$$
The <series> is absolutely convergent for every finite real $s$; in particular the expression is finite for all $s>1$, as needed for the later tail estimate.