What can we say about $a_n$ if $\sum a_n/n$ converges?

We can say that $$\frac{1}{n}\sum_{k=1}^{n}a_k = \text{AM}(a_1,\ldots,a_n) \to 0. $$ (We don't even need the hypotesis $a_n\geq 0$). This is known as Kronecker's lemma and it is a consequence of summation by parts.