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convergence sum(1,∞)1/nf(n) for f:N->N

Source: VJIMC 2010 1.1

June 3, 2021
SummationConvergence

Problem Statement

a) Is it true that for every bijection f:N→Nf:\mathbb N\to\mathbb N the series ∑n=1∞1nf(n)\sum_{n=1}^\infty\frac1{nf(n)}is convergent? b) Prove that there exists a bijection f:N→Nf:\mathbb N\to\mathbb N such that the series ∑n=1∞1n+f(n)\sum_{n=1}^\infty\frac1{n+f(n)}is convergent. (N\mathbb N is the set of all positive integers.)
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