This is related to the the May 16 post, but takes only the prime indexed terms. Does it still diverge?
Hint
Transform the product into a sum
Hint
The harmonic series 1 + 1/2 + 1/3 + ... 1/n +... diverges
this post was submitted on 31 May 2024
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Solution
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.