# Kullback-Leibler distance calculation for discrete distributions?

I have the following model

$$N \sim Pois(\lambda) \\ n \sim Bin(N,p)$$

for which I calculate the posterior for the parameter $$N$$ as

$$\pi(N|n,p,\lambda) = \frac{f(n|N,p)\pi(N|\lambda)}{f(n|p,\lambda)}$$

My goal is to calculate the Kullback-Leibler distance between the posterior and prior, i.e. $$KL = \sum \pi(N|n,p)log\frac{\pi(N|n,p,\lambda)}{\pi(N|\lambda)}$$

$$1)$$The first step is to calculate the prior predictive $$f(n|p,\lambda)$$

$$f(n|p,\lambda) = \sum_{N=0}^{\infty} \binom{N}{n}p^{n}(1-p)^{N-n}\frac{\lambda^{N}e^{-\lambda}}{N!}$$