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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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.