I have the following expression:
$$ \frac{\frac{\bar{N}^N}{(N-n)!}(1-q)^{N-n}}{\bar{N}^n\sum_{k=0}^\infty \frac{\bar{N}^k}{k!}(1-q)^k}$$
where $n,N,\bar{N}$ are positive integers, $n\leq N$, and $0\leq q\leq 1$.
I'm trying to simplify this, but I'm having a hard time getting anywhere. It took a lot of manipulation to get to this point, and I was hoping that I would be able to make use of some power series, since the sum looks a lot like the power series for an exponential function, except for the lack of relationship between $\bar{N}$ and $q$.
Does anyone have any advice? If there's something obvious that I'm missing, I'd prefer a hint, since this is a homework problem.
You were right, the sum is indeed the exponential function: $$ \sum_{k=0}^\infty \frac{\bar{N}^k}{k!}(1-q)^k = \sum_{k=0}^\infty \frac{(\bar{N}(1-q))^k}{k!} = e^{\bar{N}(1-q)}. $$