Las ecuaciones que siempre conducen a la siguiente:
\begin{align}
\sum_{n=0}^{\infty} G_{n} \, t^{n} &= e^{- \lambda} \, \sum_{n=0}^{\infty} \frac{(\lambda \, t)^{n}}{n!} = e^{-\lambda \, (1 - t)}.
\end{align}
Ahora,
\begin{align}
0 &= \sum_{n=0}^{\infty} (n+1) \left(P_{n+1} - P_{n} \right) \, t^{n} + \sum_{n=0}^{\infty} \, \sum_{k=0}^{n} G_{k} \, P_{n-k} \, t^{n} \\
&= \sum_{n=0}^{\infty} (n+1) \left(P_{n+1} - P_{n} \right) \, t^{n} + \sum_{n=0}^{\infty} \, \sum_{k=0}^{\infty} G_{k} \, P_{n} \, t^{n+k} \\
&= \frac{1}{t} \, \sum_{n=0}^{\infty} n P_{n} \, t^{n} - \sum_{n=0}^{\infty} n \, P_{n} \, t^{n} - P(t) + e^{-\lambda (1-t)} \, P(t) \\
&= \frac{1-t}{t} \, \sum_{n=0}^{\infty} n \, P_{n} \, t^{n} - (1 - e^{-\lambda(1-t)}) \, P(t)
\end{align}
esto se convierte en
\begin{align}
\frac{\sum_{n=0}^{\infty} n \, P_{n} \, t^{n}}{P(t)} = \frac{t \, \left( 1 - e^{-\lambda (1-t)}\right)}{1-t}
\end{align}
donde $P(t) = \sum_{n=0}^{\infty} P_{n} \, t^{n}$. Tomando el límite de $t \to 1$ el resultado
\begin{align}
\lim_{t \to 1} \left\{ \frac{\sum_{n=0}^{\infty} n \, P_{n} \, t^{n}}{\sum_{n=0}^{\infty} P_{n} \, t^{n}} \right\} = \lim_{t \to 1} \left\{ \frac{t \, \left( 1 - e^{-\lambda (1-t)}\right)}{1-t} \right\}
\end{align}
conduce a
\begin{align}
\sum_{n=0}^{\infty} n \, P_{n} = \lambda
\end{align}
cual es el resultado esperado.