This problem is taken from Vojtěch Jarník International Mathematical Competition 2010, Category I, Problem 1. — edit by KennyTM
On going through this post Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges? i happened to get the following 2 problems into my mind:
Let $f: \mathbb{N} \to \mathbb{N}$ be a bijection. Then does the series $$\sum\limits_{n=1}^{\infty} \frac{1}{nf(n)}$$ converge?
Next, consider the series $$\sum\limits_{n=1}^{\infty} \frac{1}{n+f(n)}$$ where $f: \mathbb{N} \to \mathbb{N}$ is a bijection. Clearly by taking $f(n)=n$ we see that the series is divergent. Does there exist a bijection such that the sum above is convergent?