I have a probability distribution of the form
$$ p_{m+1}(s)= \frac {(bs)^m}{b(m!)} e^{-bs}$$
I want to show that under the limit $m \to \infty$, it will becomes a Gaussian.
I applied Stirling's formula on the factorial but I can't massage the expression into the form I want. Can someone please help me out?