Reciprocals of Prime-Primes

Calculus Level 2

Let \(P_k\) be the \(k^\text{th}\) prime: \(P_1 = 2,\) \(P_2 = 3,\) and so on. As it turns out, \(\displaystyle \sum_{k=1}^{\infty} \frac{1}{P_k}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5} + \frac{1}{7} + \cdots\) diverges.

Does \( \displaystyle \sum_{k=1}^{\infty} \frac{1}{P_{P_k}} = \frac{1}{3}+\frac{1}{5}+\frac{1}{11}+\frac{1}{17} + \cdots\) converge?