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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.

edited Dec 24 2011 at 1:05

Goldstern
5,260●1●12●27

asked Dec 15 2011 at 10:31

Mircea
937●4●13

1

I know that there is no slowest rate of decay for convergent series, in the sense that there is no sequence $a_n$ of positive real numbers such that $\sum_n a_n |b_n| < \infty$ if and only if ${b_n}$ is a bounded sequence. This seems related to your question, but I don't think it answers it. – Paul Siegel Dec 15 2011 at 11:54

Let $a_n$ be a convergent series. We can consider series of the form $b_n = a_{f(n)}/g(f(n))$ where $g(n)$ is a sequence that increases really fast and $f(n)$ increases correspondingly slowly. If we choose $f$ and $g$ correctly then $\sum_{i=1}^n b_n \leq \sum_{i=1}^{f(n)} a_n$ for all $a_n$. A minimal element would have to be stable under all such transforms of this type. – Will Sawin Dec 24 2011 at 3:25

More simply, let $s_n$ decrease to $0$, then there is always an $a_n$ so that $\sum a_ns_n$ is convergent but $\sum a_n$ is divergent. It seems unlikely that you could therefore have any good notion of minimality. – Will Sawin Dec 24 2011 at 3:30

Juris Steprans · Answer 1 · 2011-12-15 11:56:24Z UTC

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.

Defining the slowest divergent series

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