Series whose $a_n \rightarrow 0$ as $n \rightarrow \infty$ and diverges

Question

I was wondering if there are series whose $a_n \rightarrow 0$ as $n \rightarrow \infty$ and diverges. I know one example, that is $a_n = \frac{1}{n}$. Slight modifications don't count, such as $a_n = \frac{c}{n}$ or $a_n = \frac{1}{n+k}$ where $c$ and $k$ belong to $o(n)$ (small o notation). These are effectively the same (at least the way I prove them). Is there a way to generalize all types of series which follow this criteria?