This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:
I know that a method of slowing a divergent series of positive reals is to replace the $n$-th term by it divided by the first $n$ terms. In this way the series obtained stays divergent, but it decreases infinitely faster.
Now consider the class $\Sigma$ of divergent series made of positive reals, where $(a_i)<(b_i)$ means that $\lim_{i\to\infty}a_i/b_i=0$. Consider now a decreasing sequence of series.
The question is if there is a notion of convergence fit to this order (most plausibly in a weak/generalized sense), and if there is an extension of $\Sigma$ where one could give some meaning to "the slowest divergent series".
I suspect that the/some answer would have to do with something like nonstandard analysis: one might then reframe even the definition of the order relation, in the natural way.. I would highly appreciate other speculations about the statement of the problem too.
Best Answer
This question was considered by du Bois-Reymond in 1870 and he came to some conclusions about convergence classes being linearly ordered that were not well substantiated. Hausdorff then considered the question of pantachies (maximal linearly ordered convergence classes) in great detail around 1908 and cleared up some of the ambiguities left by du Bois-Reymond. Hardy also considered this question from a different perspective. In modern terms the cardinal invariants b and d play a crtitcal role.
However, the upshot is, I believe, that no notion of least divergence class leads to any interesting concept. Non standard analysis will not help much here. For example, non-standard models obtained as ultrapowers will linearly order the convergence classes but their cofinality is non-trivial and leads to an interesting field of study.