The probability generating function of a non-negative, integer valued random variable $A$ is given by:
$G(b) = \cfrac{e^{2(b-1)}}{2-b}, (|b| \lt 2)$
To determine $\mathrm{\boldsymbol{\mathbb{E}}}(A)$ we take $G'(1)$, now I have done this, but I had to ask, what is the point of the $(|b| \lt 2)$ part?
This is most likely related to the convergence of the Geometric series: if $b \geq 2$ or $b \leq -2$, then $\frac{1}{2-b} = \frac{1}{2(1-\frac{b}{2})} \neq \frac{1}{2} \cdot \sum_{k=0}^{\infty}(\frac{b}{2})^k$