Geometric Moments via the Probability Generating Function

Let $X \sim \text{Geometric}(p)$ with $P(X = k) = (1-p)^{k-1} p$ for $k = 1, 2, 3, \ldots$

(a) Derive the probability generating function $G_X(s) = E[s^X]$ in closed form for $|s| < \frac{1}{1-p}$.

(b) Using $G_X$, compute $E[X]$ and $\text{Var}(X)$.

Recall: $E[X] = G_X'(1)$ and $\text{Var}(X) = G_X''(1) + G_X'(1) - [G_X'(1)]^2$.