I'm reading a proof where they conclude that
$$\sum_{k=1}^{\infty} \frac{O(\ln k)}{\epsilon^2\ln^2k} = \sum_{k=1}^{\infty}o_k(1) = \infty,$$
for a $\epsilon > 0$. Here the subscript $k$ means that it is dependent on another function $f_k$.
I understand that a function $g \in o(1)$ means that it converges to $0$. But I guess I'm having a hard time interpreting what the summation means, i.e. does
$$\sum_{k=1}^{\infty}o_k(1) = \infty$$
mean the summation of functions that converge to $0$ is actually not finite? How so?