Partition generation

Calculus

Let the partition function $P(n)$ enumerate the ways $n$ can be expressed as a distinct sum of positive integers, e.g. $P(4) = 5$ since $4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1$ are the only ways to represent $4$ .

$\prod_{p \ \text{prime}} \left[ \sum_{n=0}^{\infty} P(n)p^{-n} \right]$

Does the above product converge?