Conditionally convergent series

Question

I am looking for the original reference discussing a specific, elementary example of a rearrangement of series converging to a value different from the original series. In what follows, I give some (meager) context and make explicit the example whose origin I am trying to locate.

Apparently, Dirichlet was first to note that a conditionally convergent series may be rearranged so that the resulting series converges to a different value. In 1837, in his paper showing that any arithmetic progression $(a+bn)_{n\ge0}$ contains infinitely many primes provided that $a,b$ are relatively prime, he gives an example and also proves that rearranging an absolutely convergent series has no effect on its limit.

In 1853, Riemann proved his rearrangement theorem, although it was not published until 1866, as part of his Habilitationsschrift on representation of functions as trigonometric series. See here for Riemann's papers.

Theorem (Riemann). Given any α≤β in the extended reals, a conditionally convergent series of reals can be rearranged so that the liminf of the partial sums of the rearranged series is α, while the limsup is β. In particular, any real can be obtained as the sum of some rearrangement of the original series.

Prior to Riemann's theorem, Ohm had obtained interesting related result, that were later extended by Schlömilch, and Pringsheim, see here and here for statements of some of their results, and modern extensions. They are usually discussed in terms of rearrangements of the alternating harmonic series, although their results are more general.

The alternating harmonic series is the series $$1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = \ln(2)$$

Ohm's theorem predicts the value of the series obtained if we fix positive integers $p$ and $q$, and the terms of the series are rearranged so that first we add the first$p$ positive terms, then the first $q$ negative terms, then the next p positive terms, then the next q negative ones, etc. For instance, if $p=1$ and $q=2$, the rearrangement is $$1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - \cdots$$

Ohm's theorem states that this series converges to $$\ln(2) + (1/2) \ln(p/q)$$ In particular, for $p=1$, $q=2$, the limit is $(1/2) \ln(2)$.

Nowadays, when discussing rearrangements (say, in calculus textbooks), it is common to find this specific example, evaluated as follows: $$1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - \cdots = (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 + \cdots = (1/2)(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + \cdots)$$

The proof of the general result is more intricate, but introductory discussions of rearrangements omit it, while the argument above is completely basic and intuitive. (The other example that tends to be found in introductory discussions is $$1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + \cdots = (3/2) \ln(2)$$

corresponding to $p=2$, $q=1$ in Ohm's theorem. This example can be verified by a similar, only slightly more involved trick, see here.)

I am trying to locate the first place where this specific proof is mentioned. Manning's classical (1906) book Irrational numbers and their representation by sequences and series mentions the example, in pages 97, 98, but the only source it mentions is "(Laurent)", without further bibliographical details.

Finally, although perhaps off topic for this list, if a similar elementary observation can be used to prove Ohm's theorem, or a large fragment of it, I would be grateful for details. The general case is not difficult, but the proofs I am aware of are not quite as suitable for discussion in an introductory course.

Answer 1

This is the second part of my "preliminary answer". Both parts together are a little more than a third of what I ultimately intend to post, but I don't know when I'll get around to doing so. However, these two parts are certainly sufficient to give the early historical development of this topic. (later) It turns out that even the portion I wanted to post goes several thousand characters over the limit, so I'm going to have to include a third part because I want to include some stuff about Paul Laurent.

[8] Georg Friedrich Bernhard Riemann (1826-1866), Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe [On the representability of a function by a trigonometric series], Habilitationsschrift Thesis (Göttingen University, under Johann Carl Friedrich Gauss, 1777-1855), December 1853.

Riemann's rearrangement result is stated and proved in the first few sentences of Article 3. Riemann's Thesis is in the form of a handwritten manuscript and it is currently in Riemann's Nachlass at the Göttingen University Library Archives. Prior to 1868, it is likely that only a few people had seen Riemann's Thesis (mainly Berlin mathematicians, and possibly some others while visiting Berlin), with perhaps a few others having heard about its contents by word of mouth. The first printed and circulated version was published in 1867, pp. 87-131 in Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867) (JFM 1.0131.03). The 1867 publication was due to the efforts Dedekind. Although Dedekind's introductory footnote on the first page is dated July 1867 and the date given on the title page is 1867, I believe that printed copies did not reach the mathematical public until 1868. A French translation by Jean Gaston Darboux (1831-1917) and Guillaume Jules Hoüel [Houël] (1823-1886) was published in 1873: Sur la possibilité de représenter une fonction par une série trigonométrique, Bulletin des Sciences Mathématiques et Astronomiques (1) 5 (1873), pp. 20-48 and 79-96 (JFM 5.0230.03). English translations were published in Baker/Christenson/Orde's 2004 book Collected Papers (pp. 219-256) and in Hawking's book God Created the Integers (pp. 826-865 in 2005 edition; pp. 992-1031 in 2007 edition).

First few sentences of Section 3, from pp. 226-227 of the 2004 English translation: The question of the representation by trigonometric series of everywhere integrable functions with finitely many maxima and minima [in a bounded interval] was first settled rigorously by Dirichlet$^{26}$ in a paper of January 1829. {{footnote: $^{26}$Crelle's Journal, vol. IV, p. 157.}} The recognition of the proper way to attack the problem came to him from the insight that infinite series fall into two distinct classes, depending on whether or not they remain convergent when all the terms are made positive. In the first class the terms can be arbitrarily rearranged; in the second, on the other hand, the value is dependent on the ordering of the terms. Indeed, if we denote the positive terms of a series in the second class by* $a_{1},$ $a_{2},$ $a_{3},$ $\ldots$, and the negative terms by $b_{1},$ $b_{2},$ $b_{3},$ $\ldots$, then it is clear that $\sum a$ as well as $\sum b$ must be infinite. For if they were both finite, the series would still be convergent after making all the signs the same. If only one were infinite, then the series would diverge. Clearly now an arbitrarily given value $C$ can be obtained by a suitable reordering of the terms. We take alternately the positive terms of the series until the sum is greater than $C,$ and then the negative terms until the sum is less than $C.$ The deviation from $C$ never amounts to more than the size of the term at the last place the signs were switched. Now, since the numbers $a$ as well as the numbers $b$ become infinitely small with increasing index, so also are the deviations from $C.$ If we proceed sufficiently far in the series, the deviation becomes arbitrarily small, that is, the series converges to $C.$ The rules for finite sums only apply to the series of the first class. Only these can be considered as the aggregates of their terms; the series of the second class cannot. This circumstance was overlooked by mathematicians of the previous century, most likely, mainly on the grounds that the series which progress by increasing powers of a variable generally (that is, excluding [certain isolated] individual values of this variable) belong to the first class. Clearly the Fourier series do not necessarily belong to the first class. The convergence cannot be derived, as Cauchy futilely attempted $[\cdots]$

It is well known that Dirichlet met with Riemann nearly daily during the 1852 autumn vacation and that much of the first three sections of Riemann's thesis was provided by Dirichlet. Thus, it seems very possible to me that the rearrangement theorem actually originated with Dirichlet or was conceived by them together during these meetings. However, I have not encountered anything that considers this possibility.

There are many minor extensions of this result that appeared later, and it might be interesting to document the various explicit appearances of these minor extensions. For example, any conditionally convergent series can be rearranged to additionally have $+\infty$ as a sum, to have $-\infty$ as a sum, to oscillate (i.e. $\liminf$ of partial sums less than $\limsup$ of partial sums), to oscillate boundedly $(-\infty < \liminf < \limsup < +\infty),$ to oscillate infinitely $(-\infty < \liminf < \limsup = +\infty$ or $-\infty = \liminf < \limsup < +\infty$ or $-\infty = \liminf < \limsup = +\infty).$ The most precise such statement is that, given any conditionally convergent series and given any extended real numbers $a$ and $b$ such that $-\infty \leq a \leq b \leq +\infty,$ there exists a rearrangement of the terms of the series (in which the original ordering of the positive terms is not changed and the ordering of the negative terms is not changed) such that the $\liminf$ of the partial sums equals $a$ and the $\limsup$ of the partial sums equals $b.$ Moreover, for each such series and for each such extended real numbers $a$ and $b,$ there exist continuum many distict rearrangements of the series having this property. (Of course, for $a=b,$ "infinitely many" is implict in Ohm's, Schlömilch's, and Pringsheim's results for the cases where they are applicable. In fact, "continuum many" in the case of Pringsheim's results.) And one could additionally ask whether continuum many such rearrangements exist each of which has the property that every real number between $a$ and $b$ is a subseries sum, or even more, whether every real number between $a$ and $b$ is a subseries sum in which the indexing of the terms of the subseries has upper density $1$ in the sequence of natural numbers, and there are other possibilities. In the references that follow I have mostly ignored these extensions because my lack of Foreign language reading knowledge would make it extremely difficult to determine EXACTLY what an author states, what an author claims to prove, and what is a reasonable (but possibly ahistorical) claim for what an author actually proves. Nonetheless, I do give discussions of such extensions in Dini (1868), Capelli/Garbieri (1886), Trachtenberg (1903), Dini (1907), and Hobson (1907).

[9] Joseph Louis François Bertrand (1822-1900), Traité d'Algèbre [Elementary Treatise on Algebra], 2nd edition, L. [Louis] Hachette et Compagnie (Paris), 1855, ii + 495 pages.

The 1851 1st edition was titled Traité Élémentaire d'Algèbre, but after the 1st edition the title was shortened by omitting the word "Élémentaire". The 1851 1st edition has a short section on infinite series (Chapter XXII, Articles 281-289, pp. 405-412), but there seems to be no mention of the possibility of obtaining different sums by rearranging the terms in an infinite series. However, rearrangements are discussed in the 1855 2nd edition in Article 202 (pp. 224-225). In Article 202 Bertrand considers the alternating harmonic series, $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots,$ and its rearrangement in which two positive terms are followed by one negative term, $1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots$ Bertrand explains (incorrectly) that if, in each series, we consider the partial sum up to $\frac{1}{2n-1},$ then the second series includes an additional $n-1$ many negative terms, namely $-\frac{1}{n},$ $-\frac{1}{n+1},\;\ldots$ $-\frac{1}{2n-2},$ and thus the positive difference between the two partial sums exceeds $n-1$ times $\frac{1}{2n-2},$ and hence the sums of the two series differ by at least $\lim\limits_{n \rightarrow \infty}\frac{n-1}{2n-2} = \frac{1}{2}.$ This conclusion is of course incorrect, since the actual difference between the sums is $\frac{3}{2}\ln 2 - \ln 2 = \frac{1}{2} \ln {2},$ which is less than $\frac{1}{2} \ln e = \frac{1}{2}.$ To correct Bertrand's discussion, we note that we want to consider the partial sums up to $\frac{1}{2n-1}$ for even values of $n$ (a minor oversight), and in doing so the the second series includes an additional $n/2$ many negative terms, namely $-\frac{1}{n},$ $-\frac{1}{n+2},\;\ldots,$ $-\frac{1}{2n-2},$ and thus the positive difference between the two partial sums exceeds $n/2$ times $\frac{1} {2n-2},$ and hence the sums of the two series differ by at least $\lim\limits_{n \rightarrow \infty}\frac{n/2}{2n-2} = \frac{1}{4}.$ The corresponding discussion for rearranging terms in an infinite series is treated differently in Part Two of the 1863 3rd edition, which was co-authored with Paul Henri Garcet (1815-1871). (Beginning with the 3rd edition, Bertrand's Traité d'Algèbre was published in two separately paged parts, which were sometimes bound together and sometimes not bound together.) In the 1863 edition of Part Two, the discussion takes place on pp. 17-19, and for the same two series the authors show by an algebraic manipulation method (same as used in Scheibner (1860) below) that the sum of the rearranged series is $\frac{3}{2}$ times the sum of the original series. Bertrand does not mention anyone (Dirichlet, Ohm, etc.) regarding this topic in any of the editions of his book I've seen. From what I've been able to determine, the corresponding discussion of rearrangements in the many later editions of Bertrand's Traité d'Algèbre seems to be unchanged from that given in the 1863 3rd edition. The 1851 1st edition is reviewed in Nouvelles Annales de Mathématiques (1) 10 (1851), pp. 154-156 (only discusses the Appendix, pp. 405-532). The 1900 17th edition is reviewed in Mathematical Gazette 2 #26 (March 1901), pp. 34-35.

[10] Otto [Oscar, Oskar] Xaver Schlömilch (1823-1901), Bemerkung über unendliche reihen [Comments about infinite series], Zeitschrift für Mathematik und Physik 1 (1856), 180-181.

This paper discusses problems with certain manipulations of conditionally convergent series, and Ohm's name seems to be mentioned in the last paragraph. The paper does not include any specific series having a rearrangement with a different sum, unless such an example is described verbally. I was not able to obtain a coherent translation of most of this paper using online translation devices, even when I typed in the entire paper for this purpose.

[11] Auguste Joseph Alphonse Gratry (1805-1872), Philosophie. Logique [Philosophy. Logic], Volume 1, 4th edition, Charles Douniol (Paris) and J. Lecoffre & Cie (Paris), 1858, cxliv + 417 pages.

There exist 5 editions, published in 1855, 185?, 1858, 1858 (above), and 1868. (By "published in", I mean the year given on the title page. I have found that many internet and digitized references give 1856 as the date of the 1st edition, while others give 1855, and no one seems to say anything about this discrepancy.) The comments below about rearranging conditionally convergent series do not appear in the 1st edition. I have not been able to examine the 2nd edition. The comments do appear in the 3rd, 4th, and 5th editions. An English translation by Helen Singer and Milton Borah Singer of the 1868 5th edition was published by The Open Court Publishing Company in 1944 (xii + 628 pages). (from pp. 65-66 of the 1944 English edition) [p. lxxix of the 3rd & 4th editions; p. 80 of the 5th edition] The following example will make what I call the abyss which separates the infinite from the finite understandable, and show why it is not possible to infer from the one to the other. It is the example of semi-convergent series, where the infinite introduces singular properties, different from those of the finite. Take in this series a finite number of terms as large as you please. Their sum will obviously always be the same in whatever order they are added. But who would believe that this same series, assumed to be infinite, takes on the inexplicable characteristic, that the sum of its terms is different when they are added in a different order? Let the series be, for example, $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7}.$ [Note: In the original 3rd, 4th, 5th French editions, ellipses $\ldots$ were used here and just below.] Assume this to be an infinite series. The sum of its terms in the order given is equal to $\log_{e}2.$ But arrange the terms as follows: $1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4}.$ Then the sum of the terms changes and becomes equal to $\frac{3}{2} \log_{e}2.$ Hence, if we wished, accordingly, to infer from the finite to the infinite; if, after having demonstrated what is obvious in advance, that the terms of the series prolonged as far as we please, but still finite, always have the same sum in whatever order the addition is made, we then wished to maintain that it should be the same when the series is supposed infinite, that would be an error. In a lengthy footnote (footnote 10 on pp. 66-67 of the English edition; footnote 1 on pp. lxxx-lxxxi of the 3rd & 4th editions; footnote 1 on pp. 81-82 of the 5th edition), Gratry observes that all the partial sums of the series $\frac{1}{4} + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \cdots$ will be fractions that (when reduced to lowest terms) have an odd numerator and an even denominator, but the sum of the corresponding infinite series is $\frac{1}{3}.$ Gratry also mentions a theorem of Cauchy's on the (Taylor) expansion of $f(x+h)$ in which exact and precise conditions are stated for it to be valid, and then he makes the following jab at his fellow philosophers: If philosophers worked in the same fashion $[\ldots]$ they also would make discoveries. After this, Gratry mentions a letter of Abel's (in his Oeuvres, II, p. 266) in which Abel makes his now well-known comment that he does not know of any infinite series whose sum has been rigorously determined. (I haven't looked at the particular letter cited, but I suspect Gratry overstates what Abel actually said, because I believe Abel said except for simple cases like geometric series. For instance, see the more complete account on pp. 1-2 of the 1843 English translation of Ohm's book above.) Review by Ernest Nagel (1901-1985) of the 1944 English translation in Journal of Philosophy 42 #21 (11 October 1945), 580-582. Review by Ray Harbaugh Dotterer (1880-1967) of the 1944 English translation in Philosophical Review 54 #6 (November 1945), 623-624.

[12] Charles Hippolyte Berger (1822-1869), Théorie Élémentaire des Séries [Elementary Theory of Series], 2nd edition, Mallet-Bachelier (Paris), 1859, iv + 5-47 pages.

The 1st edition (which I have not seen a copy of) is 24 pages and titled Leçons sur les Séries [Lectures on Series]. The 1st edition appeared in December 1858 and I believe it existed only in lithographic form and had very limited circulation, mainly at the Academy of Montpellier. The two examples Dirichlet gave in 1837 are discussed on pp. 33-35. No mathematician's name is mentioned in this discussion, although several mathematicians (but not Dirichlet) are mentioned in the Avertissement [Preface] on pp. iii-iv. Berger first discusses the rearrangement of the alternating "square root" harmonic series in which two positive terms are followed by one negative term, showing divergence by making a direct comparison (after some algebraic manipulation) with the series whose $n$'th term is $\frac{\sqrt{2} - 1}{\sqrt{2} \sqrt{n} + 1}.$ Berger obtains the divergence of this latter series by using is Sixième Théorème on p. 19, a result that is sometimes called the logarithmic test and which is due to Cauchy (1821). However, Berger then follows this with a remark (on p. 34) in which he shows that the divergence can also be obtained by making an appropriate direct comparison with (a positive multiple of) the harmonic series. Berger then discusses the rearrangement of the alternating harmonic series in which two positive terms are followed by one negative term, showing that the rearranged series converges and that its sum is different from the sum of the alternating harmonic series. Later (on p. 38), Berger happens to mention that the sum of the alternating harmonic series is $\ln {2}$ (a result that I do not believe Berger obtains anywhere in his book) but, as far as I can tell, Berger does not mention that the rearranged alternating harmonic series has sum $\frac{3}{2}\ln {2},$ nor does his discussions concerning the rearranged alternating harmonic series allow one to (easily) conclude that its sum is $\frac{3}{2}$ times the sum of the alternating harmonic series.

[13] Eugène Charles Catalan (1814-1894), Traité Élémentaire des Séries [Elementary Treatise on Series], Leiber and Faraguet (Paris), 1860, viii + 132 pages.

The Avant-Propos [Introduction] is dated 15 February 1860. I believe this may be the earliest textbook entirely devoted to infinite series. (Stern's 1860 Lehrbuch der Algebraischen Analysis is arguably a contender for this. Incidentally, Stern's book looks like it would say something about rearrangement of series somewhere, but I've been through it page-by-page twice and can find nothing that appears to be such a discussion. However, because I cannot read German at all, there may have been a verbal discussion on rearrangement of terms that I overlooked.) Chapter VII begins with Article 197 (p. 121) in which Catalan introduces the idea of transforming slowly converging alternating series in order to speed up their convergence for computational purposes. In this discussion Catalan includes the following footnote [translated from French]: "$(^{*})$ Not content to solve this problem, several geometers have claimed to transform certain divergent series into convergent series. We believe that this statement is nonsense." Since Cauchy, Dirichlet, and Ohm would probably not be called "geometers", I wonder if the reference could be to Riemann, or perhaps to some Italian mathematicians as a result of Cauchy's early 1830s stay in Turin. Or it might have something to do with the 1858 visits by the Italian mathematicians Betti, Brioschi, and Casorati that I mention in the discussion of Novi's 1863 book below. Thus far I have not seen any mention in the published literature about this comment by Catalan. However, at the bottom of p. 250 of Bertrand (1864) (his calculus text; see below) there is also a comment about "geometers". Review by Otto [Oscar, Oskar] Xaver Schlömilch (1823-1901) [select "Read Online", then page indicator 530/570] in Zeitschrift für Mathematik und Physik 5 (1860), Literaturzeitung [Literary supplement], p. 75 (in German; separately paged). Another review is in Heidelberger Jahrbücher der Literatur 54 (1861), 121-126 (in German). The last paragraph of Schlömilch's review follows [translated from German]: Some key questions remain unanswered, for example under what circumstances is the sum of a series a continuous function and whether and under what conditions the [term-by-term] differentiation of a series is allowed, etc.; also lacks the theory of double series, which plays an important role, especially in transformations [of series].

[14] Wilhelm Scheibner (1826-1908), Über Unendliche Reihen und deren Convergenz [On Infinite Series and Their Convergence], Verlag von S. Hirzel (Leipzig), 1860, 48 pages.

The date 20 September 1860 appears on p. 3. In a footnote on p. 10 Scheibner makes reference to p. 48 of Dirichlet's 1837 paper. On p. 10 Scheibner reminds the reader that the alternating harmonic series has sum $\ln {2},$ and then he shows by an algebraic manipulation method that, by rearranging terms in the alternating harmonic series so that two positive terms are followed by one negative term, the resulting sum is $\frac{3}{2}$ times the sum of the original series. Scheibner's derivation of this result makes use of an algebraic manipulation method that reappears in a nearly identical way in many of the later textbook references that follow, such as the two 1862 editions of Schlömilch's books, p. 251 of Bertrand (1864), pp. 289-290 of Niewenglowski (1889), pp. 103-104 of Harkness/Morley (1898), pp. 21-22 of Whittaker (1902), pp. 45-46 of Godefroy (1903), pp. 96-97 of Manning (1906). The algebraic "bookkeeping work" is presented in a slightly different (but equivalent form) in Harkness/Morley, Whittaker, and Manning, but in these other references the details are almost exactly those on p. 10 of Scheibner's work. Scheibner's work is the earliest publication I know of that gives this algebraic manipulation method. Review in Heidelberger Jahrbücher der Literatur 53 (1860), pp. 823-828 (in German). In this review, the topic of rearranging terms in an infinite series is mentioned at the bottom of p. 825. For some biographical and bibliographical information about Scheibner, see Poggendorff (1863, M-Z, middle of p. 783; 1898, M-Z, middle left column p. 1181). Curiously, neither of the Poggendorff entries mentions this 1860 publication by Scheibner.

[15] Schlömilch, Handbuch der Algebraischen Analysis [Handbook of Algebraic Analysis], 3rd edition, Friedrich Frommann (Jena, Germany), 1862, viii + 414 pages.

There exist 6 editions, published in 1845, 1851, 1862 (above), 1868, 1873, and 1881 (reprinted in 1889). The topic of rearranging terms in an infinite series to obtain a different sum does not seem to appear in the 1845 1st edition or in the 1851 2nd edition. In the 1862 3rd edition this topic is discussed at the beginning of §29. Bedingte und unbedingte Convergenz (pp. 116-117). Schlömilch shows by an algebraic manipulation method (same as used on p. 10 of Scheibner (1860)) that, by rearranging terms in the alternating harmonic series so that two positive terms are followed by one negative term, the resulting sum is $\frac{3}{2}$ times the sum of the original series. Schlömilch does not mention anyone (Dirichlet, Ohm, etc.) regarding this topic. I have not seen the 4th or 5th editions, so I do not know how the treatment of this topic might be different in these two editions. In the 1881/1889 6th edition this topic is discussed a little more completely in §30. Bedingte und unbedingte Convergenz (pp. 121-123). Among other additions, there is a footnote that mentions Dirichlet's 1837 paper and the result in Schlömilch (1872) and Schlömilch (1873) (see below). Review by Georg Wilhelm Strauch (1811-1868) in Heidelberger Jahrbücher der Literatur 38 (1845), pp. 889-915 (in German). Review by Guillaume Jules Hoüel (1823-1886) in Nouvelles Annales de Mathématiques (2) 3 (1864), pp. 512-518 (in French). Review in Zeitschrift für Mathematik und Physik 10 (1865), Literaturzeitung [Literary Supplement], p. 36 (in German and French; separately paged; not an independent review--refers the reader to Hoüel's 1864 review and gives an excerpt from Hoüel's review).

[16] Schlömilch, Compendium der Höheren Analysis [Compendium of Higher Analysis], Friedrich Vieweg und Sohn, 1862, xii + 559 pages.

There exist 5 editions, published in 1853, 1862 (above), 1868, 1874, and 1881. In the 1853 1st edition infinite series are discussed in Chapter VIII (pp. 151-162), but I can find no discussion in the 1853 edition about the possibility of rearranging the terms of an infinite series to obtain a different sum. In the 1862 2nd edition (on pp. 180-181) Schlömilch shows by an algebraic manipulation method (same as used on p. 10 of Scheibner (1860)) that, by rearranging terms in the alternating harmonic series so that two positive terms are followed by one negative term, the resulting sum is $\frac{3}{2}$ times the sum of the original series. Schlömilch's discussion of this is nearly identical to the discussion he gives on pp. 116-117 of the 1862 3rd edition of his book Handbuch der Algebraischen Analysis. His discussion in Compendium ... appears to be unchanged (identical wording and identical math expressions) in all the later editions, and no person's name (Dirichlet, Ohm, etc.) is mentioned regarding this topic. Review by Johann Albert Arndt (1811-1882) in Archiv der Mathematik und Physik (1) 20 (1853), pp. 977-988 and 999-1005 (in German). Review by Schlömilch in Zeitschrift für Mathematik und Physik 8 (1863), Literaturzeitung [Literary Supplement], pp. 27-28 (in German; separately paged; not a review--a reprint of the 2nd edition preface). Review by Guillaume Jules Hoüel (1823-1886) in Nouvelles Annales de Mathématiques (2) 9 (1870), pp. 385-392 (in French; follow-up comments by Louis Philippe Gilbert (1832-1892) in NAM (2) 11, 1872, pp. 217-221).

Answer 2

See my two 7-Feb-2022 comments above. Given how long it's likely to be before I complete this project and the very incomplete details about the result one can find (internet or publications), I thought it might be best to offer at least some of the "annotated chronologically-ordered bibliography" now, and revise it as appropriate at some later time. What follows does not include some preliminary comments I intended to make, because they are not in very good form now (mostly just notes to myself about certain things), and does not go up to the late 1910s (after WW I is where a natural stopping point seems to be, but I'm stopping well before that for now).

Regarding my italization and other conventions when giving excerpts from publications (includes translations of excerpts), see Conventions near the beginning of this answer.

[1] Augustin Louis Cauchy (1789-1857), Résumés Analytiques [Analytical Summaries], l'Académie des sciences de Turin [Accademia delle Scienze di Torino; Academy of Sciences of Turin] (Italy), 1833, 166 pages.

Reprinted on pp. 7-182 in Cauchy, Œuvres Complètes, série 2, tome 10, Gauthier-Villars et fils, 1895, 184 pages. A table of contents, not in the 1833 publication, is given on pp. 183-184 of the 1895 publication. The following is from Grattan-Guinness (1990, Convolutions in French Mathematics, 1800-1840, Volume II, p. 1228): Cauchy gave his own vision of mathematical analysis further airing when in 1833 he published his Résumés analytiques in Turin (1833c), a 166-page summary of the Cours d'analyse and certain related papers. Many of the basic definitions and results were repeated more or less verbatim: not only his definition of infinitesimals but also Theorem 1136.1 [G-G's numbering; appears on p. 721] on the continuity of the sum-function, convergence tests, series expansions of basic functions, trigonometry, complex variables, de Moivre's theorem, and so on. In his foreword he promised a series of livaisons [= installments] to continue upon this book, but they did not appear. The earliest published observation I know of that the sum of an infinite series can be affected by a rearrangement of its terms appears near the beginning of §8. Des séries doubles ou multiples. Nombres de Bernoulli (pp. 56-68). Specifically, Cauchy gives an example of a rearrangement of the alternating harmonic series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ that results in a series that does not converge.

(from middle of p. 58 to top p. 59) [translated from French] But if, instead of adding the terms to each other, $$\mp \, \frac{1}{n+1}\,, \;\;\; \pm \, \frac{1}{n+2}\,, \;\;\; \mp \, \frac{1}{n+3}\,, \;\;\; \text{etc.} \;\;\; .....$$ taken in the order they are located, we change this order by choosing among the terms assigned the same sign, for example, the following $$\pm \, \frac{1}{n+2}\,, \;\;\; \pm \, \frac{1}{n+4}\,, \;\;\; ..... \;\;\; \pm \, \frac{1}{n+2n} \; = \; \pm \, \frac{1}{3n}\,,$$ [then] the numerical value of the sum of these terms, as follows, $$\frac{1}{n+2} \; + \; \frac{1}{n+4} \; + \; ..... \; + \; \frac{1}{3n}$$ obviously exceeds the product $$n \times \frac{1}{3n} \; = \; \frac{1}{3}\,,$$ and ceases to be infinitely small for infinitely large values of $n.$ After this, Cauchy returns to discussing "multiple series" (double series, triple series, etc. whose terms are defined by the use of double subscripts, triple subscripts, etc.), saying that when all the terms are positive then, by a theorem previously stated, it follows that the convergence (and sum?) is not affected by various methods of adding the terms.

The rearrangement Cauchy briefly described has the form $1 - \frac{1}{2} + a_1 - a_2 + a_3 - \cdots + (-1)^{n+1}a_n + \cdots,$ where $a_1 = \frac{1}{3}$ and $a_2 = \frac{1}{4} + \frac{1}{6}$ and $a_3 = \frac{1}{5} + \frac{1}{7} + \frac{1}{9}$ and so on, where each $a_n$ represents a finite sequence of same-signed terms from the original sequence such that the ordering of these same-signed terms among themselves is not altered and where the number of these same-signed terms is such that this number times the absolute value of the last term included equals $\frac{1}{3}.$ For example, $a_4 = \frac{1}{8} + \cdots + \frac{1}{18}$ and $a_5 = \frac{1}{11} + \cdots + \frac{1}{27}.$ Note that $a_4$ includes $6$ terms and $6(\frac{1}{18}) = \frac{1}{3},$ while $a_5$ includes $9$ terms and $9(\frac{1}{27}) = \frac{1}{3}.$ By design, $a_n \geq \frac{1}{3}$ for each $n,$ so $1 - \frac{1}{2} + a_1 - a_2 + a_3 - \cdots + (-1)^{n+1}a_n + \cdots$ diverges by the $n$-term divergence test (the $n$th term doesn't approach zero). Thus, the sequence of partial sums of this series does not converge, and since this sequence of partial sums is a subsequence of the sequence of partial sums of the rearranged series, it follows that the rearranged series diverges.

(from pp. 157-158 of Bottazzini/Gray's 2013 Hidden Harmony ...) Cauchy taught in Turin from October 1832 to July 1833, when he was called to Prague to serve as a tutor to the son of the exiled king Charles X. In accordance with the title of his chair, he delivered introductory lectures of a philosophical-theological character (Cauchy 1888a). The essential content of his Turin lectures was published in the Résumés analytiques (Cauchy 1833b) which appeared in print when he was already living in Bohemia, and contain nothing new by comparison to his previous treatises. A vivid account of Cauchy's teaching was provided by Luigi Federico Menabrea.$^{text{38}}$ Not surprisingly, Cauchy's lectures [begin centered quote] were very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius (quoted in Belhoste 1991, 156). [end centered quote] so much so that Menabrea was the only one to "see it through" out of some 30 students who initially enrolled in the course.

[2] Johann Peter Gustav Lejeune Dirichlet (1805-1859), Beweis des satzes, dafs [= dass] jede unbegrenzte arithmetische progression, deren erstes glied und differenz ganze zahlen ohne gemeinschaftlichen factor sind, unendlich viele primzahlen enthält [Proof of the theorem, that every unlimited arithmetic progression whose first member and difference are whole numbers without a common factor, contains infinitely many prime numbers], Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin. Mathematische, aus dem Jahre 1837 [for the year 1837], published in 1839, pp. 45-71.

Another digital copy is here Reprinted on pp. 315-345 of G. Lejeune Dirichlet's Werke, Volume 1, 1889. For an English translation by Ralf Stephan, see here. A statement appearing just above the first line of the paper says (translated): Read to the Akademie der Wissenschaften on 27 July 1837. Note: Page 71 is incorrectly labeled as "81". This typo has led to some incorrect "45-81" page ranges having been published: Joseph Dauben's 1979 book Georg Cantor ... (p. 368, line -6), Jesper Lützen's 1990 book Joseph Liouville ... (p. 824, line 12), Morris Kline's 1990 Volume 3 of Mathematical Thought ... (p. 830, footnote 35; p. 966, footnote 54), Bottazzini/Gray's 2013 Hidden Harmony ... (p. 782, 3rd item from top), versions 1 and 2 of Stephan's English translation (footnote on first page). Dirichlet makes some comments about the convergence of infinite series beginning near the bottom of p. 48 and continuing onto p. 49. What follows is excerpted from pp. 3-4 of version 2 (latest version; 24 November 2014) of Stephan's English translation: Before we go on it is necessary to state the reason for the condition made above, that $s>1$ should hold. We can convince ourselves of the necessity of this limitation if we respect the essential difference which exists between two kinds of infinite series. $[\cdots]$ So, for example, of the two series made from the same terms $$ 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \frac{1}{\sqrt{5}} - \frac{1}{\sqrt{6}} + \cdots,$$ $$ 1 + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{5}} + \frac{1}{\sqrt{7}} - \frac{1}{\sqrt{4}} + \cdots,$$ only the first converges while of the following $$ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots,$$ $$ 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots,$$ both converge, but with different sums. Our infinite series $L,$ as can be easily seen, belongs only then to the first of the classes $[\cdots]$ I cannot find any explicit mention of Cauchy in this paper.

[3] Dirichlet, Démonstration de cette proposition: toute progression arithmétique dont le premier terme et la raison sont des entiers sans diviseur commun, contient une infinité de nombres premiers [Proof of this proposition: every arithmetic progression such that the first term and common ratio are integers without common divisor, contains an infinity of prime numbers], Journal de Mathématiques Pures et Appliquées [= Liouville's Journal] (1) 4 (1839), 393-422.

A footnote (by Joseph Liouville, 1809-1882) at the bottom of p. 393 states that this is a translation by Olry Terquem (1782-1862) of Dirichlet's 1837 paper. Dirichlet's remarks about rearranging terms of an infinite sequence extend from the bottom of p. 396 to the middle of p. 397.

[3] Dirichlet, Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres [Investigations on diverse applications of infinitesimal analysis to the theory of numbers], Journal für die Reine und Angewandte Mathematik 19 #4 (1839), 324-369.

Dirichlet gives 4 July 1839 (on p. 369) for the date of the paper's completion. Dirichlet makes some comments about the convergence of infinite series beginning on p. 329 (line -10) and continuing onto p. 330, where I believe Dirichlet discusses the fact that rearrangements of certain series can lead to different values. However, I do not see any specific examples given. It might be the case that a specific example is described verbally, though. I cannot find any explicit mention of Cauchy in this paper.

[4] Martin [Marcin, Martinus] Ohm (1792-1872), **De Nonnullis Seriebus Infinitis Summandis** [Concerning the Summation of Certain Infinite Series], Trowitzschii et Filii [Trowitzsch und Sohn; Trowitzsch and Son] (Berlin), 1839, 15 pages.

A 300+ page biography of Ohm was published in 1987, but I have not been able to examine it. This is a separately published booklet, written in Latin, that deals with the summation of certain rearrangements of the alternating harmonic series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.$ In §1 (p. 3) the alternating harmonic series is shown to have sum $\ln {2}.$ Ohm's verification consists of plugging $x=0$ into the Maclaurin series for $\ln (1+x).$ The remaining verifications are similar in that Ohm first obtains series expansions of certain rational functions and their natural logarithms. In §2 (pp. 3-6) the rearrangement in which $2$ positive terms are followed by $1$ negative term is shown to have sum $\frac{3}{2}\ln {2}.$ In §3 (pp. 6-7) the result for $1$ positive term followed by $2$ negative terms is shown to have sum $\frac{1}{2}\ln {2}.$ In §4 (pp. 7-8) the result for $3$ positive terms followed by $1$ negative term is shown to have sum $\frac{1}{2}\ln {3}.$ In §5 (pp. 8-9) the result for $3$ positive terms followed by $2$ negative terms is shown to have sum $\frac{1}{2}\ln 2 + \frac{1}{2}\ln {3}.$ In §6 (pp. 9-11) the result for $m$ positive terms followed by $1$ negative term is shown to have sum $\ln 2 + \frac{1}{2}\ln {m}.$ In §7 (pp. 11-12) the result for $1$ positive term followed by $n$ negative terms is shown to have sum $\ln 2 - \frac{1}{2}\ln {n}.$ In §8 (pp. 12-14) the result for $m$ positive terms followed by $n$ negative terms is shown to have sum $\ln 2 + \frac{1}{2}\ln \frac{m}{n}.$ Finally, §9 (p. 14) gives a summary of the result proved in §8, and §10 (pp. 14-15) appears to involve some type of concluding remarks. The only reference cited is in a footnote on p. 14, which directs the reader to see Aufsätze aus dem Gebiete der höhern Mathematik (Berlin, 1823), and no name appears anywhere except on the title page (in particular, neither Cauchy nor Dirichlet is mentioned). Some comments about Ohm and infinite series (but not the results he published in 1839) are given on pp. 30-31 of Cajori's 1888 paper History of infinite series.

[5] Ohm, Der Geist der Mathematischen Analysis und ihr Verhältniss zur Schule [The Spirit of Mathematical Analysis and Its Relationship to the Field], Erste Abhandlung [First Treatise], Duncker und Humblot (Berlin), 1842, xvi + 159 pages.

An English translation by Alexander John Ellis (1814-1890), titled The Spirit of Mathematical Analysis, and Its Relation to a Logical System, was published in 1843. After p. 103 of the English translation there are 3 pages (not numbered) that give some information about Ohm's published books. Ohm's 1839 essay De Nonnullis Seriebus Infinitis Summandis is not included. Incidentally, I highly recommend reading the Introduction (by Ohm) of the Ellis translation for an excellent account of many of the problems mathematicians were having at this time with interpreting various formal manipulations. The following is "[observation] (2)" on p. 56 [= pp. 87-88 of the 1842 German]: Now a convergent series has this value only by virtue of the law according to which its terms proceed in infinitum. Hence to prevent any doubt occurring concerning the true value of a convergent series, we must carefully enunciate the law according to which the terms are to be taken in infinitum. For example, from the same reciprocal terms of the natural numbers, viz. $$1, \; -\frac{1}{2}, \; +\frac{1}{3}, \; -\frac{1}{4}, \; +\frac{1}{5}, \; -\frac{1}{6}, \; +\frac{1}{7}, \; - \,\text{in inf.}$$ we may compose any number of numerical infinite series, which are all convergent but have all different values, each however having its own perfectly determinate value by virtue of the determinate law according to which it is constructed. The series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + \, \text{in inf.}$$ in which, if we take $2n$ terms, there are always as many positive as negative terms, has for its value log. nat. $2.$ The series $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6} + \, \text{in inf.}$$ in which, if we take the first $3n$ terms, $2n$ positive and only $n$ negative of the above terms follow one another (in their order) has for its value $\frac{3}{2}\,$log. nat. $2.$ The series $$1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \frac{1}{12} + \frac{1}{7} - \, \text{in inf.}$$ in which, if we take the first $3n$ terms, we find $n$ of them positive and $2n$ negative, has for its value $\frac{1}{2}\,$log. nat. $2.$ And if we take $\mu$ of the above terms positive and $\nu$ negative, and then $\mu$ positive and $\nu$ negative, and so on, the value of the resulting numerical convergent series is $=\,$log. nat. $2 + \frac{1}{2}\,$log. nat.$\,\frac{\mu}{\nu},$ and therefore $=\,$log. nat. $2,$ which is that of the first series, when $\mu = \nu,$ but greater than that when $\mu > \nu$ and less when $\mu < \nu.$ For an outline of the method Ohm uses, see the top half of p. 818 of Cowen/Davidson/Kaufman (1980) below. Ohm is not mentioned in this 1980 paper, however.

[6] Victor Amédée Lebesgue (1791-1875), Sur la convergence des séries [On the convergence of series], Nouvelles Annales de Mathématiques (1) 4 (1845), 66-70.

For some biographical and bibliographical information about V. Lebesgue, see Poggendorff (1898, A-L, middle left column p. 784 to bottom left column p. 785). There appears to be no family/blood relationship between V. Lebesgue and Henri Lebesgue (known for the Lebesgue integral). (from bottom of p. 69 to end of the paper) [translated from French; math notation slightly modernized in some places] Proposition V. In a series containing negative terms, if we change the order of the terms, [then] we can $1^{\text o}$ change the sum without destroying the convergence; $2^{\text o}$ destroy the convergence, that is to say, render the series divergent instead of convergent. [Proof of] $1^{\text o}$ The series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{2n+1} - \frac{1}{2n+2},$$ [and] $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots + \frac{1}{4n+1} + \frac{1}{4n+3} - \frac{1}{2n+2},$$ are both convergent, because in the first the terms are decreasing, and it will be the same in the second by replacing $1 + \frac{1}{3}$ by $\frac{4}{3},$ [and] $\frac{1}{5} + \frac{1}{7}$ by $\frac{12}{35},$ etc. Moreover, if we take for general terms $\frac{1}{2n+1} - \frac{1}{2n+2} = \frac{1}{(2n+1)(2n+2)};$ [and] $\frac{1}{4n+1} + \frac{1}{4n+3} - \frac{1}{2n+2} = \frac{8n+5}{(4n+1)(4n+3)(2n+2)},$ [then] each term in the second series exceeds the corresponding term in the first, [and thus] we conclude that the sum of the first series is less than that of the second. [Proof of] $2^{\text o}$ The series $1 - \sqrt{\frac{1}{2}} + \sqrt{\frac{1}{3}} - \sqrt{\frac{1}{4}} + \cdots$ is convergent; but if instead of taking the negative terms $2$ by $2,$ we take them as above, $3$ by $3,$ [then] the series $1 + \sqrt{\frac{1}{3}} - \sqrt{\frac{1}{2}},\;$ $\sqrt{\frac{1}{5}} + \sqrt{\frac{1}{7}} - \sqrt{\frac{1}{4}},\;$ $\ldots,\;$ $\sqrt{\frac{1}{4n+1}} + \sqrt{\frac{1}{4n+3}} - \sqrt{\frac{1}{2n+2}}\;$ [a typo has been corrected], will be divergent. Cauchy's rule gives, assuming $n$ is very large, $\;\sqrt{\frac{1}{4n+1}} + \sqrt{\frac{1}{4n+3}} - \sqrt{\frac{1}{2n+2}} \; = \; \sqrt{\frac{1}{n}} - \sqrt{\frac{1}{2n}} \; = \; \frac{{\sqrt 2} - 1}{{\sqrt 2}{\sqrt n}},\;$ and $\;\frac{\ln \left(\frac{{\sqrt n}{\sqrt 2}}{{\sqrt 2} - 1}\right)}{\ln n} \; = \; \frac{1}{2} + \frac{\ln \left(\frac{\sqrt 2}{{\sqrt 2} - 1}\right)}{\ln n} \; < \; 1.$ These examples are taken from a memoir of Mr. Dirichlet, where it is established that any arithmetic progression contains infinitely many prime numbers, when the first term and the common difference are relatively prime.

[7] Charles-Adrien Choquet (1798-1880) and Mathias Mayer [Mayer-d'Almbert] (1786-1843), Traité Élémentaire d'Algèbre [Elementary Treatise on Algebra], 5th edition, Bachelier (Paris), 1849, xvi + 638 pages.

There exist 5 editions, published in 1832 (infinite series on pp. 198-201; nothing about rearrangements changing sums), 1836 (infinite series on pp. 244-247 and pp. 528-532; nothing about rearrangements changing sums), 1841 (infinite series on pp. 260-264 and pp. 578-583; nothing about rearrangements changing sums), 1845 (infinite series on pp. 260-264 and pp. 578-583; nothing about rearrangements changing sums), and [1849] (above; series on pp. 244-248 and pp. 583-588 and pp. 634-638; the possibility of rearrangements changing sums is discussed on pp. 634-638). In the first four editions the ordering of the authors on the title page differs from that of the 1849 5th edition in that in these earlier editions "Mayer" is listed first and "Choquet" is listed second. Because Mayer died in 1843, the 1845 4th edition and 1849 5th edition were by Choquet only. I do not know whether any corrections and additions to the last two editions may have been influenced by possible notes left by Mayer on suggestions for later editions, but I suspect the additional comments in the 5th edition about rearranging terms in the alternating harmonic series are entirely due to Choquet. Following the 1849 5th edition Choquet published two editions of Complément d'Algèbre (intended as a supplement to the 1849 5th edition), the 1st edition in 1851 (iv + 51 pages) and the 2nd edition in 1853 (iv + 63 pages), but neither of these includes any general discussions about infinite series. Finally, in 1856 Choquet published Traité d'Algèbre [Treatise on Algebra], Mallet-Bachelier (Paris), 1856, xvi + 551 pages. On pp. 548-551 of this 1856 book, Choquet discusses rearranging terms of a series. The 1856 version appears to be the same (word-for-word) as the 1849 version. Review of 5th edition in Nouvelles Annales de Mathématiques (1) 8 (1849), pp. 429-431 (in French). Some information about Choquet and Mayer and their books can be found here under the entry for "Urbain Le Verrier".

What follows is a translation of pp. 636-638 of the 1849 5th edition. 3. In order to show that, when convergence occurs only as a result of the signs, rearrangement of the terms can change the sum or render the series divergent, let us consider the series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + ....$$ Let us take in succession two positive terms and one negative, in the following way $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + ....$$ This new series can be formed from the first, by adding to it that which results from dividing all its terms by $2.$ Thus, if one distributes the terms in groups of four terms, the $n^{\text{th}}$ group is $$\frac{1}{4n-3} - \frac{1}{4n-2} + \frac{1}{4n-1} - \frac{1}{4n};$$ after the division of all the terms by $2,$ [and then] grouping each term with its successor, the $n^{\text{th}}$ group is $$\frac{1}{4n-2} - \frac{1}{4n};$$ the sum of these two groups is $$\frac{1}{4n-3} + \frac{1}{4n-1} - \frac{1}{2n},$$ and if in succession [we use] $n=1,$ $=2,$ $=3,$ etc., the result is, successively, the groups of three terms of the series $1 + \frac{1}{3} - \frac{1}{2} +$ *etc. [Note: I will give a more explicit description. In what follows, $n$ is a positive integer (i.e. $n=1,$ $2,$ $3,\;\ldots).$ Let Series A be the series whose $n$'th term is the sum of the $(4n-3)$'th and $(4n-2)$'th and $(4n-1)$'th and $4n$'th terms of the first series (the alternating harmonic series). Let Series B be the series whose $n$'th term is $\frac{1}{2}$ times the $n$'th term of the first series. Let Series C be the series whose $n$'th term is the sum of the $(2n-1)$'th and $2n$'th terms of Series B. Let Series D be the series whose $n$'th term is the sum of the $n$'th term of Series A and the $n$'th term of Series C. Then the $n$'th term of Series D is $\frac{1}{4n-3} + \frac{1}{4n-1} - \frac{1}{2n},$ and this value is also equal to the sum of the $(3n-2)$'th and $(3n-2)$'th and $3n$'th terms of the new series (the rearranged alternating harmonic series).] Let $s_{2n}$ and $s_{4n}$ be the sum of the first $2n$ and of the first $4n$ terms of the first series; that of the first $2n$ terms of the series $\frac{1}{2} - \frac{1}{4} + \frac{1}{6} - \frac{1}{8} +$ etc. will be $\frac{1}{2}s_{2n};$ and the sum of the first $3n$ terms of the series obtained by rearrangement of the terms will be $s_{4n} + \frac{1}{2}s_{2n}.$ But, if we indefinitely increase the number $n,$ [then] the sums $s_{4n}$ and $s_{2n}$ converge to the same limit, which is $l\,2$ $[= {\ln 2}]$ $(n^{\text{o}}\;534)$ [= (article $534)].$ Therefore, $s_{4n} + \frac{1}{2}s_{2n}$ converges to $\frac{3}{2}l\,2.$ Thus the series $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots$$ has sum $\frac{3}{2}l\,2.$ 4. By another change in the order of the terms of the same series, we can obtain a divergent series. This would be if we alternately took groups of positive terms, then of negative terms, in which the number of terms increased by stages and indefinitely, in a manner that the value of each group was always greater than a given number, as the group below, for example, $$\frac{1}{n} + \frac{1}{n+2} + \frac{1}{n+4} + \cdots + \frac{1}{3n-2},$$ whose value is greater than $\frac{n}{3n-2}$ and therefore greater than $\frac{1}{3}.$ 5. By considering the convergent series $$1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \cdots,$$ we can even more easily derive a divergent series. It suffices to alternately take two positive terms then a negative term, as follows: $$1 + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{5}} + \frac{1}{\sqrt{7}} - \frac{1}{\sqrt{4}} + ....$$ Indeed, if we put [In the first equation below have included a plus sign just after the term $-\frac{1}{\sqrt{4}}$ that did not appear in the original.] $$f(n) \;\; = \;\; 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \cdots - \frac{1}{\sqrt{2n-2}} + \frac{1}{\sqrt{2n-1}} - \frac{1}{\sqrt{2n}},$$ $$F(n) \;\; = \;\; 1 + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} + \cdots - \frac{1}{\sqrt{2n-2}} + \frac{1}{\sqrt{4n-3}} + \frac{1}{\sqrt{4n-1}} - \frac{1}{2n},$$ we have $$F(n) - f(n) \;=\; \frac{1}{\sqrt{2n+1}} + \frac{1}{\sqrt{2n+3}} + \cdots + \frac{1}{\sqrt{4n-1}}.$$ The second member of this equality is greater than $\frac{n}{\sqrt{4n-1}}$ and therefore greater than $\frac{1}{2}\sqrt{n}.$ It therefore tends to infinity along with $n.$ Therefore it is the same for $F(n)-f(n),$ and hence, for $F(n),$ since $f(n)$ is a finite quantity.

Answer 3

This is the third part of my "preliminary answer". At the end of this part we're only up to 1867, although what I've done (which needs a few additions I've found in the last several years, along with my intended re-proofing of everything) goes up to about 1919, and thus the majority of the material is not included in these three parts.

[17] Paul Mathieu Hermann Laurent (1841-1908), Théorie des Séries [Theory of Series], Mallet-Bachelier (Paris), 1862, viii + 124 pages.

Note: P. M. H. Laurent is not the Laurent that the Laurent expansions in complex analysis are named after. This other Laurent is Pierre Alphonse Laurent (1813-1854), and there appears to be no family/blood relationship between the two Laurents. The table of contents is on pp. 123-124. Laurent knew of Catalan's 1860 book -- see the discussion on p. 6 of Laurent's book. Laurent discusses problems with rearranging terms of the alternating harmonic series on pp. 15-19, and problems with the Cauchy product of conditionally convergent series on pp. 28-29. The example Laurent considers on pp. 15-16 is the same as the first rearrangement example in V. Lebesgue's 1845 paper, but Laurent provides a few more details. This same example using the same method (in which the rearranged series is shown to have a different sum by showing that the difference between their partial sums is bounded away from zero) reappears in Laurent's 1867 text (pp. 286-287), in Serret's 1868 text (pp. 143-145), and in Osgood's 1897 text (p. 43). Incidentally, there is an error in Laurent's general representation of the 3-grouped terms $\left(\frac{1}{1} + \frac{1}{3} - \frac{1}{2}\right),$ $\left(\frac{1}{5} + \frac{1}{7} - \frac{1}{4}\right),$ etc. Laurent uses $\left(\frac{1}{4n-1} + \frac{1}{4n+1} - \frac{1}{2n}\right),$ which does not correctly give the 3-grouped terms for $n = 1,\,2,\,3,\,\ldots.$ A correct representation for $n = 1,\,2,\,3,\,\ldots$ is $\left(\frac{1}{4n-3} + \frac{1}{4n-1} - \frac{1}{2n}\right),$ and a correct representation for $n = 0,\,1,\,2,\,\ldots$ is $\left(\frac{1}{4n+1} + \frac{1}{4n+3} - \frac{1}{2n+2}\right)$ (what V. Lebesgue used). Correcting this error involves some minor subsequent changes (e.g. $n \times \frac{1}{4n+1}$ changes to $n \times \frac{1}{4n-1}$ in the 2nd line after equation (3)), but the error itself does not affect the correctness of Laurent's conclusion that the two sums differ by at least $\frac{1}{4}.$ For some biographical and bibliographical information about P. M. H. Laurent, see Poggendorff (1898, A-L, middle right column p. 780 to top left column p. 781) and see this 25 November 2014 manuscript. Review by Pierre Marie Eugène Prouhet (1817-1867) in Nouvelles Annales de Mathématiques (2) 2 (1863), pp. 25-31 (in French).

[18] Schlömilch, Ueber die bedingt convergirenden reihen [On conditionally convergent series], Zeitschrift für Mathematik und Physik 7 (1862), 283-284.

The paper does not give an author's name. However, Schlömilch's authorship can be verified from an index entry on p. iii at the beginning of the journal volume (see first page of Inhalt, under the listings for Arithmetik und Analysis) and from the last paper listed for 1862 on p. 274 of Moritz Cantor's 1901 Nachruf an Oskar Schlömilch. Schlömilch begins by saying that originating with a remark by Dirichlet (no reference is given), we know that we are (in general) not allowed to arbitrarily rearrange the terms in an infinite series. Schlömilch gives the two examples that Dirichlet gave in his 1837 paper. For the rearrangement in which two positive terms are followed by one negative term, Schlömilch simply states that one converges to $\ln 2$ and the other converges to $\frac{3}{2}\ln 2.$ For the other example (same as previous example except that square roots are present on the terms), Schlömilch gives an elegant argument--nicer than the argument in V. Lebesgue's 1845 paper, in my opinion--for the divergence of the rearranged series. This elegant argument is repeated in Schlömilch (1881, top half of p. 31).

[19] Giovanni [Jean] Novi (1827-1866), Trattato di Algebra Superiore [Treatise of Higher Algebra], Parte Prima. Analisi Algebrica [First Part. Algebraic Analysis], Le Monnier (Firenze [Florence], Italy), 1863, viii + 458 pages.

This was intended to be the first volume of a planned 3-volume treatise, based on a manuscript (presumably handwritten) for lectures given by Enrico Betti (1823-1892) during 1858-1859, but only this first volume was ever published. The two examples Dirichlet gave in 1837 are discussed on pp. 70-71. No mathematician's name is mentioned here, but in a footnote on p. 73 Novi cites Dirchlet's 1837 paper for the result that all rearrangements of an absolutely convergent series of complex numbers have the same sum. For Dirichlet's alternating harmonic series example in which the rearrangement is where two positive terms are followed by one negative term, Novi uses the standard algebraic manipulation method to show that the sum of the rearranged series is $\frac{3}{2}$ times the sum of the original series. In the case of the corresponding rearrangement of the alternating "square root" harmonic series, Novi shows by an elementary argument that the rearranged series diverges to $+\infty.$ Regarding Betti, I have looked through some of his publications and made some internet searches (but I did not put a lot of effort into this) and I have not found any explicit discussion by him about series rearrangements having different sums. However, I do know that during the autumn 1858 vacation the Italian mathematicians Betti, Brioschi, Casorati all visited universities in Göttingen, Berlin, Paris and met Dirichlet, Riemann, Weierstrass, Bertrand, and others. See p. 468 here and the bottom of p. 529 of Baker/Christenson/Orde (2004) (see the entry for Riemann's trigonometric series thesis above). Perhaps Betti learned about rearranging conditionally convergent series at this time, which took place just before (or soon after) Betti began giving his lectures on which Novi's book is based. For some additional information about Novi's book and Betti's algebra course, see p. 204 of Laura Martini, *The first lectures in Italy on Galois theory: Bologna, 1886-1887, Historia Mathematica 26 #3 (August 1999), 201-223. Review in in Nouvelles Annales de Mathématiques (2) 3 (1864), pp. 90-91.

[20] Joseph Louis François Bertrand (1822-1900), Traité de Calcul Différentiel et de Calcul Intégral. Première Partie. Calcul Différentiel [Treatise on Differential and Integral Calculus. First Part. Differential Calculus], Gauthier-Villars, 1864, xliv + 780 pages. JFM 2.0298.01

The Preface is rather long (44 pages, on pp. i-xliv). The topic of rearranging terms of a conditionally convergent series is discussed in Article 248 (pp. 250-251). Bertrand mentions (bottom p. 250) that [translated from French] "Dirichlet, who was the first to call this point to the attention of the geometers, gave the following example:" Bertrand's example is the alternating harmonic series and he shows that rearranging the terms so that two positive terms are followed by one negative term gives a sum that is $3/2$ times the original sum. Bertrand's derivation of this result is nearly identical to the derivation on p. 10 of Scheibner (1860). For some biographical and bibliographical information about Bertrand, see Poggendorff (1898, A-L, middle left column p. 121 to top right column p. 122). [Review by Pierre Marie Eugène Prouhet (1817-1867)] (http://www.numdam.org/numdam-bin/item?id=NAM_1864_2_3__550_0) in Nouvelles Annales de Mathématiques (2) 3 (1864), pp. 550-558 (in French). Review by Joseph Dienger (1818-1894) in Heidelberger Jahrbücher der Literatur 58 (1865), pp. 903-912 (in German).

[21] Laurent, Traité d'Algèbre [Treatise on Algebra], Gauthier-Villars, 1867, xvi + 520 pages.

There exist at least 5 editions of "Part 2" (where rearrangement of series appears), published in 1867 (above), 1875, 1881, 1887, and 1897 (have not seen). Beginning with the 3rd edition, "Part 2" is separately paged and sometimes first published in a different year than "Part 1". For example, the 3rd edition of "Part 1" was first published in 1879 and the 3rd edition of "Part 2" was first published in 1881, while I believe the 5th edition of both "Part 1" and "Part 2" were first published in 1897. In the 1867 1st edition a remark on pp. 286-287 discusses the rearrangement of the alternating harmonic series in which two positive terms are followed by one negative term. The discussion is virtually word-for-word identical to the corresponding discussion on pp. 15-16 of Laurent (1862), including Laurent's earlier error in describing the general 3-grouped term. Indeed, the only changes I could detect when quickly comparing them side-by-side was a difference in wording in the first sentence (lines 3-4 of Remarque) and a colon just before equation (3) in the 1862 book that does not appear in the 1867 book. The corresponding remark in later editions seems virtually unchanged, except the error in describing the general 3-grouped term (the error appears in the 1867 1st edition and in the 1875 2nd edition) is corrected beginning with the 1881 3rd edition. Brief review in Nouvelles Annales de Mathématiques (2) 14 (1875), p. 477 (in French).

[22] Johann Lieblein (1834-1881), Sammlung von Aufgaben aus der Algebraischen Analysis [Collection of Exercises in Algebraic Analysis], H. Carl J. Satow (Prague), 1867, viii + 192 pages.

Preface dated April 1867. This is a book of student exercises designed to be used with Schlömilch's Handbuch der Algebraischen Analysis. Exercises 253-256 (pp. 46-47) give several examples of series rearrangements in which the student is asked to determine whether the rearrangement converges and, if it does, to find its sum. One example is the rearrangement of the alternating harmonic series in which three positive terms are followed by two negative terms. Another example is the corresponding series (three positive terms followed by two negative terms) in which the terms are the sine of the harmonic series terms. That is, the series $$\sin 1$ + \sin \frac{1}{3} + \sin \frac{1}{5} - \sin \frac{1}{2} - \sin \frac{1}{4} + \sin \frac{1}{7} + \sin \frac{1}{9} + \sin \frac{1}{11} - \cdots $$ The 2nd edition, titled J. Lieblein's Sammlung von Aufgaben aus der Algebraischen Analysis zum Selbstunterricht [J. Lieblein's Collection of Exercises in Algebraic Analysis for Self Study] and prepared by Wenzel Johann [= Jan Václav?] Láska (1862-1943), was published in 1889 (viii + 180 pages; JFM 20.0253.03). In the 1889 2nd edition the same examples appear as Exercises 202-205 (pp. 43-45), and at the end of Exercise 202 (rearrangements of the alternating harmonic series) references are given to p. 333 of Stern's Lehrbuch der Algebraischen Analysis (no year is given, and the only edition I've seen is the 1860 edition, which does not have any rearrangements on p. 333), to p. 172 of Schlömilch's's Uebungsbuch zum Studium der Hoeheren Analysis, and to Dirichlet's 1836 paper. Also, in the 1889 2nd edition, Exercise 201 on p. 43 sketches out the apparent "paradox" discussed at the beginning of Glaisher's 1872 paper, and Glaisher's paper (but not his name) is cited at the end of Exercise 201. Review of the 1867 edition by Guillaume Jules Hoüel (1823-1886) in Bulletin des Sciences Mathématiques et Astronomiques (1) 1 (1870), pp. 362-363 (in French).

[23] Raffaele Giovanni Rubini (1817-1890), Complemento agli Elementi d'Algebra [To Complement to the Elements of Algebra], 2nd edition, A. Morelli (Napoli [Naples], Italy), 1867, 4 + iv + 484 + iii pages.

Preface dated October 1867. I believe the 1st edition was published in 1861. I have not been able to examine a copy of the 1st edition, so it is possible that Rubini's book should appear earlier in my list. On pp. 384-385 Rubini shows that the alternating harmonic series and the rearranged version in which two positive terms are followed by one negative term have sums that differ by an amount between $\frac{1}{4}$ and $\frac{1}{2}.$ No mathematician's name is mentioned in this particular discussion, but Dirichlet's name is mentioned further down on p. 385 for the result (which Rubini gives a proof of) that all rearrangements of an absolutely convergent series of complex numbers have the same sum. Several more editions exist (a total of 5? the last in 1878?), but I have not been able to confirm this because I am not sure whether some similarly titled books by Rubini (that I've seen notices of in various journals from this time period) should be considered later editions of this book or different books. A similar difficulty existed for the books by Choquet and Mayer (see the "1849" entry above), but I was able to resolve it because I was able to examine all the books involved. The only other book by Rubini I've seen that includes something about rearrangements changing sums is Rubini's Trattato d'Algebra [Treatise on Algebra], Parte Seconda. Complemento agli Elementi d'Algebra [Part Two. To Complement the Elements of Algebra], A. Morelli (Napoli [Naples], Italy), 1873, vi + 363 pages. In this 1873 book the same discussion described above appears repeated word-for-word on p. 315. Review of the 1873 edition by Guillaume Jules Hoüel (1823-1886) in Bulletin des Sciences Mathématiques et Astronomiques (1) 6 (1874), p. 21 (in French).