I recently thought of this problem (though I by no means think others haven't...) and a couple of solutions; I figured I would share. I think this is an interesting problem for those learning analysis.
Prove or provide a counterexample: Let $\{x_n\}$ be a sequence in $\mathbb{R}$ that converges to $0$. Then $\sum_{n=0}^{\infty} \frac{x_n}{n}$ is a convergent series.
I think this is interesting because it tackles a boundary case of "how quickly must a series go to 0", as for any $\epsilon >0$, $\sum_{n=0}^{\infty} \frac{x_n}{n^{1+\epsilon}}$ converges. This is a nice situation of "can you find a sequence that converges slowly enough."
I'd also be interested to see other counter-examples.