For those interested, here is the most elementary proof that sets of the natural numbers of positive upper density necessarily have divergent reciprocal sums, r.e. Pete's discussion above.

Suppose $A \subset \mathbf{N}$ and $\displaystyle\limsup_{N \to \infty} \frac{|A\cap [1,N]|}{N} = \alpha > 0$. 

Let $N_0=1$, and choose a sequence $N_k$ such that $N_k \geq \frac{4N_{k-1}}{\alpha}$ and $|A \cap [1,N_k)| \geq \frac{\alpha}{2}N_k$ for all $k \in \mathbf{N}$. $$\sum_{n \in A}\frac{1}{n}= \sum_{k=1}^{\infty}\sum_{n \in A\cap [N_{k-1},N_k)}\frac{1}{n} \geq \sum_{k=1}^{\infty}(|A\cap [1,N_k)|-N_{k-1})\frac{1}{N_k}$$ $$ \geq \sum_{k=1}^{\infty}(\frac{\alpha}{2}N_k - \frac{\alpha}{4}N_k) \frac{1}{N_k}=\sum_{k=1}^{\infty}\frac{\alpha}{4} \to \infty.$$ This is really just a generalization of the classical proof of the divergence of the harmonic series, where you group together progressively larger collections of consecutive terms that add to at least one-half. Hope this helps!