Reciprocals of Prime-Primes

Calculus Level 2

Let PkP_k be the kthk^\text{th} prime: P1=2,P_1 = 2, P2=3,P_2 = 3, and so on. As it turns out, k=11Pk=12+13+15+17+\displaystyle \sum_{k=1}^{\infty} \frac{1}{P_k}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5} + \frac{1}{7} + \cdots diverges.

Does k=11PPk=13+15+111+117+ \displaystyle \sum_{k=1}^{\infty} \frac{1}{P_{P_k}} = \frac{1}{3}+\frac{1}{5}+\frac{1}{11}+\frac{1}{17} + \cdots converge?

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