I don't know how to begin proving it, but the more I run this series out, bigger it gets. The conditions of the equation are such that it will always have a consistently non-zero rate of increase, even though that rate of increase decreases each time the formula is cycled ((pn/pn-1) will always be more than (pn+1/pn+1-1), nonetheless any and every (pn/pn-1) will be >1). The divergence will be glacial, but definite.
Perhaps surprisingly, that's actually good enough since the sum of the prime reciprocals also diverges. However, I'm not letting you just assume that, and proving it is harder than the original problem.
What I like about this solution is that it doubles as a proof that there are infinitely many prime numbers. It's not circular either -- it uses the number theoretic fact that every integer has a prime factorization, but nothing deeper. If only finitely many primes were required for that, ... well then of course every finite product converges.