$A \subset \mathbb N$, $ \ \sum_{n\in A} \frac{1}{n}$ converges $\implies \frac{1}{n} $ #$(A \cap[1,n]) \to_{\infty} 0 $
If $A\subseteq\mathbb N$ and $\sum\limits_{a\in A}\frac1a$ converges then $A$ has natural density $0$