This one is for fun. I'll give my solution later.

Let $\displaystyle S$ be the set of squarefree positive integers. Show, as simply as you, can that $\displaystyle \sum_{s \in S} \frac{1}{s} = \infty$.

Of course an instant solution is given by the fact that the sum of the reciprocals of the primes diverges. However try using another way for fun!