Let $a_i$ be a positive sequence such that $a_i \to 0$.
I know that the series $\sum_{i=1}^\infty a_i$ may be divergent.
But what about the series divided by $n$; does the following go to 0?
$$\frac{\sum_{i=1}^n a_i}{n} \to^{n\to \infty} 0$$
In Wikipedia there are many convergence tests, but here the question is not exactly about convergence of series, so I don't know where to start...