Questions about conditionally convergent series and rearrangement of

Question

According to Riemann Series Theorem or Riemann Rearrangement Theorem a conditionally convergent series - with a clever rearrangement of terms - can converge to any desired value, or even can be shown to diverge.

Admittedly the questions below may be a bit confusing. See comments by Dave below. He articulates in different words what I was seeking.

Q1. Is there a favored value or sequence (e.g $\eta(\frac{1}{2})$ has a conditionally convergent series, but it is also defined to have value $0.6048986434216303702472...$ see here. Why should this value be given “superior among equals” status? Perhaps this has something to do with order of terms off Dirichlet series?

Q2. Suppose there are two distinct arrangements producing same result because the sequence of partial sums ${A_n}$ and ${B_n}$ both converge to the same infinite sum $S$ (Because $B_n$ Is obtained by rearranging all or most terms of $A_n$ and the difference between the partial sums goes to zero as number of terms in the partial sum goes to infinity) - is such rearrangement allowed to be swapped? Has this kind of thing done in any proof before? If yes, where. In most cases I see that people don’t want to rearrange the terms at all because rearrangement could lead to different answer. Would it always lead to different answer, or it would be allowed if certain additional conditions are satisfied by the rearrangement?

Thanks in advance.

Answer 1

A very similar question was asked on mathoverflow, Conditional convergence and rearrangements, and what follows are some more details regarding my comment there (which only mentioned Borel's paper as being relevant). Although I think Borel’s 1890 paper [3] gives the first general result, there were probably several specific results proved previously (likely little known then, even to Borel), one of which I know of is [1].

[1] Otto [Oscar, Oskar] Xaver Schlömilch (1823-1901), Ueber bedingt-convergirende reihen [On conditionally convergent series], Berichte über die Verhandlungen der Königlich SächsischenGesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Classe 24 (1872), 327-330.

Schlömilch proves the following generalization of Ohm's 1839 result. Let $u_0 - u_1 + u_2 - u_3 + \cdots$ be a convergent alternating series with sum $s.$ Assume that $\lim\limits_{n \rightarrow \infty}nu_n$ exists finitely or infinitely. [I do not know whether Schlömilch explicitly states that this limit exists, but it is not difficult to see that this limit need not exist finitely or infinitely. For example, this limit does not exist finitely or infinitely for the convergent alternating series $1 - \frac{1}{\sqrt{2}} + \frac{1}{3} - \frac{1}{\sqrt{4}} + \frac{1}{5} - \frac{1}{\sqrt{6}} + \cdots\,]$ Then the rearrangement of this series in which $p$ positive terms are followed by $q$ negative terms is $s \; + \; \frac{1}{2}\left(\lim\limits_{n \rightarrow \infty}nu_n\right)\cdot\ln \frac{p}{q}.$ Schlömilch gives two applications. The first application is that every "fixed $p$ and $q$ rearrangement" of the series $1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \cdots$ diverges to $\infty.$ The second application is that every "fixed $p$ and $q$ rearrangement" of the series $1 - \frac{1}{2\ln 2} + \frac{1}{3\ln 3} - \frac{1}{4\ln 4} + \cdots$ converges to the same sum as the original series. A third application, which I don't think Schlömilch explicitly gives, is that if $\lim\limits_{n \rightarrow \infty}nu_n$ exists finitely, then every "fixed $p$ and $q$ rearrangement" such that $p=q$ has the same sum as the original series.

[2] Otto [Oscar, Oskar] Xaver Schlömilch (1823-1901), Ueber bedingt-convergirende reihen [On conditionally convergent series], Zeitschrift für Mathematik und Physik 18 (1873), 520-522.

The paper appears to be word-for-word identical to Schlömilch's 1872 paper above. In three or four places I was able to find changes in punctuation (e.g. semi-colon instead of a colon, semi-colon removed, etc.). The paper is discussed in [5] (bottom of p. 643 to bottom of p. 644). For a short French summary of Schlömilch’s paper, see Bulletin des Sciences Mathématiques et Astronomiques (1) 6 (1874), p. 251.

[3] Émile Félix Édouard Justin Borel (1871-1956), Sur le changement de l'ordre des termes d'une série semi-convergente [On changing the order of terms in a semi-convergent series], Bulletin des Sciences Mathématiques (2) 14 (1890), 97-102.

This is among of the first two or three papers published by Borel, who published over 300 papers and books. The paper was written and published while Borel was a 1st year undergraduate (the 1889-1890 academic year). Borel's paper is cited as reference for Example 7 of Chapter IV, on p. 71, of Bromwich's 1908 book [identically worded as Example 11 on p. 77 of the 1925 2nd edition]. Bromwich's Example 7 states: In order that the value of a non-absolutely convergent series may remain unaltered after a certain change in the order of the terms, it is sufficient that the product of the displacement of the $n$th term by the greatest subsequent term may tend to zero as $n$ increases to $\infty.$ Borel's 1890 paper is also discussed in [5] (bottom of p. 644 to the end of paper).

[4] Maurice René Fréchet (1878-1973), Sur le résultat du changement de l'ordre des termes dans une série [On the result of changing the order of terms in a series], Nouvelles Annales de Mathématiques (4) 3 (1903), 507-511.

This is the 4th of over 300 papers published by Fréchet. Since Fréchet completed his undergraduate work in 1903, this paper was probably written during Fréchet's last year as an undergraduate. I haven’t tried to determine how the results in Fréchet’s paper relate to those in Borel’s paper. However, although Fréchet does not cite Borel’s paper, Fréchet surely knew about it. For instance, Fréchet essentially wrote the first 100 pages of Borel’s 1905 book Leçons sur les Fonctions de Variables Réelles et les Développements en Séries de Polynomes, which was based on lectures Borel gave during the 1903-04 winter term and which were attended by Fréchet.

[5] Gina Aurello [later: Gina Aurello Marzo], On the rearrangement of infinite series, Pi Mu Epsilon Journal 9 #10 (Spring 1994), 641-646.

(from p. 641) The purpose of this paper is to trace the origin and development of some of the results on rearranging series from Cauchy through Dirichlet and Riemann. Also, we will discuss two theorems, of O. Schlömilch and E. Borel, that are not generally known and that partially answer the natural questions of how the rearrangement of its terms affects the sum of an infinite series and for which series rearranging terms does not change the sum.

[6] Roman Wituła, Edyta Hetmaniok, and Damian Słota, On Commutation Properties of the composition relation of convergent and divergent permutations (Part I), Tatra Mountains Mathematical Publications 58 #1 (2014), 13-22.

This paper and its references is a good place to begin looking for work done since Borel’s 1890 paper [3]. There is no mention Fréchet’s paper, however. I don’t know whether this is because the authors were not aware of Fréchet’s paper, or because Fréchet’s paper is not sufficiently related to what the authors study.

Answer 2

1. Cause that's the value it converges to. Permutations of this series can converge to other values (or diverge) according to the theorem, but who cares about them? Those are permutations of the series, not the series itself. Of course it has to do with the order of the terms in the series... the whole point of the rearrangement theorem is that the order matters (a lot!).

2. I'm not sure what you mean by 'allowed to be swapped'. If two different permutations of the same series converge to the same value, then... they converge to the same value. The theorem says you can't expect this to always be the case (unless the convergence is absolute), but in general you can always find distinct permutations that converge to the same value. For instance, it's easy to see that just swapping the first two terms will not change the value of the series. For instance if we are looking at $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots\\ -\frac{1}{2}+1+\frac{1}{3}-\frac{1}{4}+\ldots,$$ the partial sums are $$ 1,\frac{1}{2},\frac{5}{6}, \frac{7}{12},\ldots\\ -\frac{1}{2},\frac{1}{2},\frac{5}{6}, \frac{7}{12}, \ldots$$ The sequences of partial sums are identical after the term, so they converge to the same thing. By the same reasoning, just permuting a finite number of terms will not change the value. Any permutation that changes the value must move an infinite number of terms.