Is there a notion of density for a (strictly increasing) sequence of natural numbers that decides whether the sum of the reciprocals of that sequence converges?

The notion of natural density gives a suficient condition. Namely, one can prove that if $A \subseteq \mathbb{N}$ has positive upper natural density then $\sum_{a \in A} \frac{1}{a}$ diverges. This condition is not a necessary one, though. A wellknown result in number theory ascertains that the series of the reciprocals of the prime numbers diverges whereas $\pi(n) = o(n)$. In the following list you are to encounter some previous discussions on MO that are closely related to the current inquiry of yours: [1] Erdos Conjecture on arithmetic progressions [2] On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... References I. Erdös's brilliant proof of the divergence of $\sum_{p} \frac{1}{p}$: http://www.renyi.hu/~p_erdos/193813.pdf II. A. Rice, Density and substance: an investigation into the size of integer subsets. 

