In building the largest co-prime numbers and dividing them into each other would the product converge or diverge?

Question

Utilizing all prime numbers and alternating which goes in the numerator and denominator to create the largest co-prime numbers on top and bottom, would the product converge or diverge?

$$3/2*7/5*13/11 ... = $$

Answer 1

The product is $$\prod\left(1+{a_p\over p}\right)\tag1$$ where the product is taken as $p$ runs through the alternate primes $2,5,11,17,\dots$ and $a_p\ge1$ is the difference between $p$ and the next prime. The product converges if and only if $$\sum{a_p\over p}\tag2$$ converges. It's well-known that $\sum p^{-1}$ diverges when the sum is taken over all primes $p$, and the same methods suffice to prove the sum diverges when taken over just every alternate prime. Since $a_p\ge1$, it follows that the sum in (2) diverges, so the product in (1) diverges.