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Let PkP_kPk be the kthk^\text{th}kth prime: P1=2,P_1 = 2,P1=2, P2=3,P_2 = 3,P2=3, and so on. As it turns out, ∑k=1∞1Pk=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} + \cdotsk=1∑∞Pk1=21+31+51+71+⋯ diverges.
Does ∑k=1∞1PPk=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} + \cdotsk=1∑∞PPk1=31+51+111+171+⋯ converge?
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