If $\sum_{k = 1}^\infty \frac{a_k}{k} < + \infty$, then $\frac{1}{n} \sum_{k = 1}^n a_k \to 0$
Slick proofs that if $\sum\limits_{k=1}^\infty \frac{a_k}{k}$ converges then $\lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^n a_k=0$