Rate of convergence of partial sums of reciprocals

Question

It is a known fact that $\left\{ \sum_{i=1}^n \frac1i \right\}$ is divergent. It is also known that for any $\epsilon > 0$, $\left\{ \sum_{i=1}^n \frac{1}{i^{1+\epsilon}} \right\}$ is convergent. What about the middle ground - sequences of the form $\sum_{i=1}^n \frac{1}{i^{1 + f(i)}}$ where $f(i)$ is small. For instance, for the choice of $f(i) = i^{-1}$, the sequence, \begin{equation*} \sum_{i=1}^n \frac{1}{i^{(1 + i^{-1})}} \end{equation*} is divergent. And so is the sequence with the choice of $f(i) = 1 / \log i$. Is there any characterization of the functions that make this sequence convergent and divergent? What about if $f (i)$ is the reciprocal of the inverse Ackermann function?