Which series converges the most slowly?

Solution 1:

There is not such a function! Walter Rudin in his "Principles of Mathematical Analysis" has a series of exercises trying to indicate that there is no function at the "boundary" between convergence and divergence.

There was an interesting series of answers about this very issue at MathOverflow (MO): https://mathoverflow.net/questions/49415/nonexistence-of-boundary-between-convergent-and-divergent-series

A quick way of seeing that there is no "slowest" convergent series is Rudin's exercise 12.b, mentioned at the link above: If $\sum a_n$ converges, and the $a_n$ are positive, then $\sum a_n/\sqrt {r_n}$ converges, where $r_n$ is the tail $\sum_{i\ge n}a_i$. Note $r_n\to 0$, so $a_n/\sqrt{r_n}>a_n$ for $n$ large enough.

I recommend that you take a look at the answers at MO and at the references they suggest, for more subtle examples.