# Infinite series with prime number [duplicate]

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, AdrianoSep 3 '13 at 22:08

• 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. – Daniel Fischer Sep 3 '13 at 21:53