I know the $\sum_{n=1}^\infty \frac{1}{\text{Prime[$n$]}}$ does not converge, but what about the following series? $$\sum_{n=1}^\infty\frac{1}{\text{Prime[Prime[$n$]]}}$$ (Where $\text{Prime[$n$]}$ is the $n$th prime number.)

I ran some calculation on Mathematica, but I am not sure in the answer gets very close to 1. Can anyone help with that?

NB: I an not a professional mathematician. I am just an amateur.


marked as duplicate by Michael Albanese, Daniel Fischer, Stefan Hamcke, apnorton, Adriano Sep 3 '13 at 22:08

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  • $\begingroup$ What is Prime[Prime[n]] ? $\endgroup$ – Salech Rubenstein Sep 3 '13 at 21:50
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    $\begingroup$ The $n$-th prime is asymptotically $n\log n$, so the $p_n$-the prime is asymptotically $(n\log n)\log (n\log n) \geqslant n(\log n)^2$, and that means the sum converges. $\endgroup$ – Daniel Fischer Sep 3 '13 at 21:53
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    $\begingroup$ Just wanted to say: even though I voted to mark as a duplicate, I think this question is a really good one--so don't take it personally or anything; finding if a question has been asked previously is very difficult on this site. $\endgroup$ – apnorton Sep 3 '13 at 22:26