The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's Principles of Mathematical Analysis, following Theorem 3.29, he writes:
One might thus be led to conjecture that there is a limiting situation of some sort, a “boundary” with all convergent series on one side, all divergent series on the other side—at least as far as series with monotonic coefficients are concerned. This notion of “boundary” is of course quite vague. The point we wish to make is this: No matter how we make this notion precise, the conjecture is false. Exercises 11(b) and 12(b) may serve as illustrations.
Exercise 11(b) states that if $\sum_n a_n$ is a divergent series of positive reals, then $\sum_n a_n/s_n$ also diverges, where $s_n = \sum_{i=1}^n a_n$. Exercise 12(b) states that if $\sum_n a_n$ is a convergent series of positive reals, then $\sum_n a_n/\sqrt{r_n}$ converges, where $r_n = \sum_{i\ge n} a_i$.
Although these two exercises are suggestive, they are not enough to convince me of Rudin’s strong claim that no matter how we make this notion precise, the conjecture is false. Are there any stronger theorems in this direction?
Edit. For example, are there any theorems about the topology/geometry of the spaces of all convergent/divergent series, where a series is viewed as a point in $\mathbb{R}^\infty$ or $(\mathbb{R}^+)^\infty$ in the obvious way?
Best Answer
A rather detailed discussion of the subject can be found in Knopp's Theory and Application of Infinite Series (see § 41, pp. 298-305). He mentiones that the idea of a possible boundary between convergent and divergent series was suggested by du Bois-Reymond. There are many negative (and mostly elementary) results showing that no such boundary, in whatever sense it might be defined, can exist.
Stieltjes observed that for an arbitary monotone decreasing sequence $(\epsilon_n)$ with the limit $0$, there exist a convergent series $\sum c_n$ and a divergent series $\sum d_n$ such that $c_n=\epsilon_nd_n$. (This can be easily deduced from the Abel-Dini theorem).
Pringsheim remarked that, for a convergent and a divergent series with positive terms, the ratio $c_n/d_n$ can assume all possible values, since one may have simultaneously $$\liminf\frac{c_n}{d_n}=0\qquad\mbox {and}\qquad\limsup\frac{c_n}{d_n}=\infty.$$
I like the following geometric interpretation. Given a (convergent or divergent) series $\sum a_n$, let's mark the sequence of points $(n,a_n)\in\mathbb R^2$ and join the consecutive points by straight segments. Then there is a convergent series $\sum c_n$ and a divergent series $\sum d_n$ (both with positive and monotonically decreasing terms) such that the corresponding polygonal graphs can intersect in an indefinite number of points.
The results remain essentially unaltered even if one requires that both sequences $(c_n)$ and $(d_n)$ are fully monotone, which is a very strong monotonicity assumption. This was shown by Hahn ("Über Reihen mit monoton abnehmenden Gliedern", Monatsh. für Math., Vol. 33 (1923), pp. 121-134).