Generating Function: Why is $G(0)=P(X=0)$?

Question

$G$ is the generating function: $$G(s)=E\left[s^X\right]=\sum_{i=0}^{\infty}P(X=i)s^i$$ But the textbook claims that $G(0)=P(X=0)$. Why? $$G(0)=\sum_{i=0}^{\infty}P(X=i)(0)^i=0$$ but this is not $P(X=0)$, this is wrong?

Answer 1

The definition for $G(s)$ and result of $G(0)$ are both correct. Since $ 0^0 = 1$, and $0^i = 0$ for $i \in \mathbb(R) \neq 0$, we have $$ G(0) = \sum_{i=0}^{\infty} P(X=i)(0)^i = P(X=i) \cdot 1 = P(X=i)$$