The sum of the reciprocal of primeth primes A few days ago, a friend of mine taught me that the sum of the reciprocal of primeth primes $$\frac{1}{3}+\frac{1}{5}+\frac{1}{11}+\frac{1}{17}+\frac{1}{31}+\cdots$$
converges.
Does anyone know some papers which have a rigorous proof with the convergence value?

It is a duplicate. We have $p_n\gg n\log n$ by Chebyshev's theorem (or the PNT) hence $$p_{p_n}\gg p_n \log n \gg n\log^2 n$$ and $$\sum_{n\geq 1}\frac{1}{n\log^2 n}$$
is convergent by Cauchy's condensation test.
