History of Grandi's series

Geometry and infinite zeros

Grandi

Guido Grandi (1671 – 1742) reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into nowrap|1=1 − 1 + 1 − 1 + · · · produced varying results: either:$(1-1)\; +\; (1-1)\; +\; cdots\; =\; 0$or:$1+(-1+1)+(-1+1)\; +cdots\; =\; 1.$

Grandi's explanation of this phenomenon became well-known for its religious overtones:

In fact, the series was not an idle subject for Grandi, and he didn't think it summed to either 0 or 1. Rather, like many mathematicians to follow, he thought the true value of the series was ¹⁄₂ for a variety of reasons.

Grandi's mathematical treatment of nowrap|1=1 − 1 + 1 − 1 + · · · occurs in his 1703 book "Quadratura circula et hyperbolae per infinitas hyperbolas geometrice exhibita". Broadly interpreting Grandi's work, he derived nowrap|1=1 − 1 + 1 − 1 + · · · = ¹⁄₂ through geometric reasoning connected with his investigation of the witch of Agnesi. Eighteenth-century mathematicians immediately translated and summarized his argument in analytical terms: for a generating circle with diameter "a", the equation of the witch "y" = "a"³/("a"² + "x"²) has the series expansion::$sum\_\{n=0\}^infty\; frac\{(-1)^nx^\{2n\{a^\{2n-1=a\; -\; frac\{x^2\}\{a\}\; +\; frac\{x^4\}\{a^3\}\; -\; frac\{x^6\}\{a^5\}\; +\; cdots$:and setting "a" = "x" = 1, one has 1 − 1 + 1 − 1 + · · · = ¹⁄₂. [According to Giovanni Ferraro (2002 p.193), citing Marco Panza's doctoral dissertation including a detailed analysis of Grandi's writing.]
*According to Morris Kline, Grandi started with the binomial expansion::$frac\{1\}\{1+x\}\; =\; 1\; -\; x\; +\; x^2\; -\; x^3\; +\; cdots$:and substituted "x" = 1 to get nowrap|1=1 − 1 + 1 − 1 + · · · = ¹⁄₂. Grandi "also argued that since the sum was both 0 and ¹⁄₂, he had proved that the world could be created out of nothing." [Kline 1983 p.307]

Grandi offered a new explanation that nowrap|1=1 − 1 + 1 − 1 + · · · = ¹⁄₂ in 1710, both in the second edition of the "Quadratura circula" [Panza (p.298) places the example on p.30 of Grandi 1710, "Quadratura circula … editio altera"] and in a new work, "De Infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica". [Reiff pp.65-66] Two brothers inherit a priceless gem from their father, whose will forbids them to sell it, so they agree that it will reside in each other's museums on alternating years. If this agreement lasts for all eternity between the brother's descendants, then the two families will each have half possession of the gem, even though it changes hands infinitely often. This argument was later criticized by Leibniz. [Leibniz (Gerhardt) pp.385-386, Markushevich p.46]

The parable of the gem is the first of two additions to the discussion of the corollary that Grandi added to the second edition. The second repeats the link between the series and the creation of the universe by God:

Marchetti

After Grandi published the second edition of the "Quadratura", his fellow countryman Alessandro Marchetti became one of his first critics. One historian charges that Marchetti was motivated more by jealousy than any other reason. [Montucla pp.8-9] Marchetti found the claim that an infinite number of zeros could add up to a finite quantity absurd, and he inferred from Grandi's treatment the danger posed by theological reasoning. The two mathematicians began attacking each other in a series of open letters; their debate was ended only by Marchetti's death in 1714.

Leibniz

With the help and encouragement of Antonio Magliabechi, Grandi sent a copy of the 1703 "Quadratura" to Leibniz, along with a letter expressing compliments and admiration for the master's work. Leibniz received and read this first edition in 1705, and he called it an unoriginal and less-advanced "attempt" at his calculus. [Mazzone and Roero pp.246-247, who cite: Grandi to Magliabechi, Pisa 17.7.1703 BU Pisa MS 99, f. 219; Magliabechi to Grandi, Florence 31.7.1703, BU Pisa MS 93, f. 110; Grandi to Leibniz, Pisa 28.6.1703, GM 4, p. 209; Leibniz to Magliabechi, Hanover 12.8.1704; Leibniz to Magliabechi, Hanover 2.7.1705, Paoli 1899, p. XC; Leibniz to Grandi, Hanover 11.7.1705, GM 4, pp.210-212; Leibniz to Hermann, Hanover 21.5.1706, GM 4, p. 297] Grandi's treatment of 1 − 1 + 1 − 1 + · · · would not catch Leibniz's attention until 1711, near the end of his life, when Christian Wolff sent him a letter on Marchetti's behalf describing the problem and asking for Leibniz's opinion. [Hitt p.141; Wolff to Leibniz, 16 April 1711, in Gerhardt pp.134-135, LXIII]

Background

As early as 1674, in a minor, lesser-known writing "De Triangulo Harmonico" on the harmonic triangle, Leibniz mentioned nowrap|1=1 − 1 + 1 − 1 + · · · very briefly in an example::$frac\{1\}\{1+1\}\; =\; frac11-frac\{1\}\{1+1\}.\; ;mathrm\{Ergo\};\; frac\{1\}\{1+1\}\; =\; 1-1+1-1+1-1\; ;mathrm\{etc.\}$ [Leibniz p.369] The series nowrap|1=1 − 1 + 1 − 1 + · · · also appears indirectly in a discussion with Tschirnhaus in 1676. [Leibniz p.817]

Leibniz had already considered the divergent alternating series nowrap|1=1 − 2 + 4 − 8 + 16 − · · · as early as 1673. In that case he argued that by subtracting either on the left or on the right, one could produce either positive or negative infinity, and therefore both answers are wrong and the whole should be finite. Two years after that, Leibniz formulated the first convergence test in the history of mathematics, the alternating series test, in which he implicitly applied the modern definition of convergence. [Leibniz pp.205-207; Knobloch pp.124-127]

olutions

In the 1710s, Leibniz described Grandi's series in his correspondence with several other mathematicians. [For example, his ultimate solution is repeated in a 1716 letter to Pierre Dangicourt; see Hitt p.143] The letter with the most lasting impact was his first reply to Wolff, which he published in the "Acta Eruditorum". In this letter, Leibniz attacked the problem from several angles.

In general, Leibniz believed that the algorithms of calculus were a form of "blind reasoning" that ultimately had to be founded upon geometrical interpretations. Therefore he agreed with Grandi that nowrap|1=1 − 1 + 1 − 1 + · · · = ¹⁄₂, claiming that the relation was well-founded because there existed a geometric demonstration. [Ferraro 2000 p.545]

On the other hand, Leibniz sharply criticized Grandi's example of the shared gem, claiming that the series nowrap|1=1 − 1 + 1 − 1 + · · · has no relation to the story. He pointed out that for any finite, even number of years, the brothers have equal possession, yet the sum of the corresponding terms of the series is zero. [Leibniz (Gerhardt) pp.385-386, Markushevich p.46]

Leibniz thought that the argument from nowrap|1=1/(1 + "x") was valid; he took it as an example of his law of continuity. Since the relation nowrap|1=1 − "x" + "x"² − "x"³ + · · · = 1/(1 + "x") holds for all "x" less than 1, it should hold for "x" equal to 1 as well. Still, Leibniz thought that one should be able to find the sum of the series nowrap|1=1 − 1 + 1 − 1 + · · · directly, without needing to refer back to the expression nowrap|1=1/(1 + "x") from which it came. This approach may seem obvious by modern standards, but it is a significant step from the point of view of the history of summing divergent series. [As noted by Weidlich (p.1)] In the 18th century, the study of series was dominated by power series, and summing a numerical series by expressing it as "f"(1) of some function's power series was thought to be the most natural strategy. [Ferraro and Panza p.32]

Leibniz begins by observing that taking an even number of terms from the series, the last term is −1 and the sum is 0::1 − 1 = 1 − 1 + 1 − 1 = 1 − 1 + 1 − 1 + 1 − 1 = 0.Taking an odd number of terms, the last term is +1 and the sum is 1::1 = 1 − 1 + 1 = 1 − 1 + 1 − 1 + 1 = 1.

Now, the infinite series 1 − 1 + 1 − 1 + · · · has neither an even nor an odd number of terms, so it produces neither 0 nor 1; by taking the series out to infinity, it becomes something between those two options. There is no more reason why the series should take one value than the other, so the theory of "probability" and the "law of justice" dictate that one should take the arithmetic mean of 0 and 1, which is nowrap|1=(0 + 1) / 2 = 1/2. [Leibniz (Gerhardt) pp.386-387; Hitt (p.143) translates the Latin into French.]

Eli Maor says of this solution, "Such a brazen, careless reasoning indeed seems incredible to us today…" [Maor, pp.32-33] Kline portrays Leibniz as more self-conscious: "Leibniz conceded that his argument was more metaphysical than mathematical, but said that there is more metaphysical truth in mathematics than is generally recognized." [Kline 1983 pp.307-308]

Charles Moore muses that Leibniz would hardly have had such confidence in his metaphysical strategy if it did not give the same result (namely ¹⁄₂) as other approaches. [Moore p.2] Mathematically, this was no accident: Leibniz's treatment would be partially jusitifed when the compatibility of averaging techniques and power series was finally proven in 1880. [Smail p.3]

Reactions

When he had first raised the question of Grandi's series to Leibniz, Wolff was inclined toward skepticism along with Marchetti. Upon reading Leibniz's reply in mid-1712, [Wolff's first reference to the letter published in the "Acta Eruditorum" appears in a letter written from Halle, Saxony-Anhalt dated 12 June 1712; Gerhardt pp.143-146.] Wolff was so pleased with the solution that he sought to extend the arithmetic mean method to more divergent series such as nowrap|1=1 − 2 + 4 − 8 + 16 − · · ·. Leibniz's intuition prevented him from straining his solution this far, and he wrote back that Wolff's idea was interesting but invalid for several reasons. For one, the terms of a summable series should decrease to zero; even nowrap|1 − 1 + 1 − 1 + · · · could be expressed as a limit of such series. [Moore's pp.2-3; Leibniz's letter is in Gerhardt pp.147-148, dated 13 July 1712 from Hanover.]

Leibniz described Grandi's series along with the general problem of convergence and divergence in letters to Nicolaus I Bernoulli in 1712 and early 1713. J. Dutka suggests that this correspondence, along with Nicolaus I Bernoulli's interest in probability, motivated him to formulate the St. Petersburg paradox, another situation involving a divergent series, in September of 1713. [Dutka p.20]

According to Pierre-Simon Laplace in his "Essai Philosophique sur les Probabilités", Grandi's series was connected with Leibniz seeing "an image of the Creation in his binary arithmetic", and thus Leibniz wrote a letter to Jesuit missionary Claudio Filippo Grimaldi, court mathematician to the Kangxi Emperor in China, in the hope that the emperor's interest in science and the mathematical "emblem of creation" might combine to convert the nation to Christianity. Laplace remarks, "I record this anecdote only to show how far the prejudices of infancy may mislead the greatest men." [Upham and Stewart pp.479, 480, who cite Laplace pp.194, 195.]

Divergence

Jacob Bernoulli

Jacob Bernoulli (1654 – 1705) dealt with a similar series in 1696 in the third part of his "Positiones arithmeticae de seriebus infinitis". Applying Nicholas Mercator's method for polynomial long division to the ratio nowrap|1="k"/("m" + "n"), he noticed that one always had a remainder. [Ferraro 2002 p.181] If nowrap|1="m" > "n" then this remainder decreases and "finally is less than any given quantity", and one has:$frac\{k\}\{m+n\}=frac\{k\}\{m\}\; -\; frac\{kn\}\{m^2\}\; +\; frac\{kn^2\}\{m^3\}\; -\; cdots.$If "m" = "n", then this equation becomes:$frac\{k\}\{2m\}=frac\{k\}\{m\}\; -\; frac\{k\}\{m\}\; +\; frac\{k\}\{m\}\; -\; cdots.$Bernoulli called this equation a "not inelegant paradox".Knopp p.457] [Cantor (p.96) makes the quote "unde paradoxum fluit non inelegans", citing Ebenda II, 751.]

Varignon

Pierre Varignon (1654 – 1722) treated Grandi's series in his report "Précautions à prendre dans l'usage des Suites ou Series infinies résultantes…". The first of his purposes for this paper was to point out the divergence of Grandi's series and expand on Jacob Bernoulli's 1696 treatment.

(Varignon's math…)

The final version of Varignon's paper is dated February 16, 1715, and it appeared in a volume of the "Mémories" of the French Academy of Sciences that was itself not published until 1718. For such a relatively late treatment of Grandi's series, it is surprising that Varignon's report does not even mention Leibniz's earlier work. [For the possible significance of the omission, see Panza p.339.] But most of the "Précautions" was written in October 1712, while Varignon was away from Paris. The Abbé Poignard's 1704 book on magic squares, "Traité des Quarrés sublimes", had become a popular subject around the Academy, and the second revised and expanded edition weighed in at 336 pages. To make the time to read the "Traité", Varignon had to escape to the countryside for nearly two months, where he wrote on the topic of Grandi's series in relative isolation. Upon returning to Paris and checking in at the Academy, Varignon soon discovered that the great Leibniz had ruled in favor of Grandi. Having been separated from his sources, Varignon still had to revise his paper by looking up and including the citation to Jacob Bernoulli. Rather than also take Leibniz's work into account, Varignon explains in a postscript to his report that the citation was the only revision he had made in Paris, and that "if" other research on the topic arose, his thoughts on it would have to wait for a future report. [Panza p.339; Varignon pp.203, 225; Gerhardt p.187]

(Letters between Varignon and Leibniz…)

In the 1751 "Encyclopédie", Jean le Rond d'Alembert echoes the view that Grandi's reasoning based on division had been refuted by Varignon in 1715. (Actually, d'Alembert attributes the problem to "Guido Ubaldus", an error that is still occasionally propagated today.) [Hitt pp.147-148]

Riccati and Bougainville

In a 1715 letter to Jacopo Riccati, Leibniz mentioned the question of Grandi's series and advertised his own solution in the "Acta Eruditorum". [Bagni (p.4) identifies the letter as "probably written in 1715", citing Michieli's 1943 "Una famiglia di matematici…", p. 579] Later, Riccati would criticize Grandi's argument in his 1754 "Saggio intorno al sistema dell'universo", saying that it causes contradictions. He argues that one could just as well write nowrap|1="n" − "n" + "n" − "n" + · · · = "n"/(1 + 1), but that this series has "the same quantity of zeroes" as Grandi's series. These zeroes lack any evanescent character of "n", as Riccati points out that the equality nowrap|1=1 − 1 = "n" − "n" is guaranteed by nowrap|1=1 + "n" = "n" + 1. He concludes that the fundamental mistake is in using a divergent series to begin with:

Another 1754 publication also criticized Grandi's series on the basis of its collapse to 0. Louis Antoine de Bougainville briefly treats the series in his acclaimed 1754 textbook "Traité du calcul intégral". He explains that a series is "true" if its sum is equal to the expression from which is expanded; otherwise it is "false". Thus Grandi's series is false because nowrap|1=1/(1 + 1) = 1/2 and yet nowrap|1=(1 − 1) + (1 − 1) + · · · = 0. [Bougainville vol.1, ch.22, pts.318-320, pp.309-312; Schubring p.29]

Euler

Leonhard Euler treats nowrap|1=1 − 1 + 1 − 1 + · · · along with other divergent series in his "De seriebus divergentibus", a 1746 paper that was read to the Academy in 1754 and published in 1760. He identifies the series as being first considered by Leibniz, and he reviews Leibniz's 1713 argument based on the series nowrap|1=1 − "a" + "a"² − "a"³ + "a"⁴ − "a"⁵ + · · ·, calling it "fairly sound reasoning", and he also mentions the even/odd median argument. Euler writes that the usual objection to the use of nowrap|1=1/(1 + "a") is that it does not equal nowrap|1=1 − "a" + "a"² − "a"³ + "a"⁴ − "a"⁵ + · · · unless "a" is less than 1; otherwise all one can say is that:$frac\{1\}\{1+a\}\; =\; 1\; -\; a\; +\; a^2\; -\; a^3\; +\; cdots\; pm\; a^n\; mp\; frac\{a^\{n+1\{1+a\},$where the last remainder term does not vanish and cannot be disregarded as "n" is taken to infinity. Still writing in the third person, Euler mentions a possible rebuttal to the objection: essentially, since an infinite series has no last term, there is no place for the remainder and it should be neglected. [Euler 1760 §§3-5, pp.206-207; English translation in Barbeau and Leah pp.145-146] After reviewing more badly divergent series like nowrap|1=1 + 2 + 4 + 8 + · · ·, where he judges his opponents to have firmer support, Euler seeks to define away the issue:

bquote|1=Yet however substantial this particular dispute seems to be, neither side can be convicted of any error by the other side, whenever the use of such series occurs in analysis, and this ought to be a strong argument that neither side is in error, but that all disagreement is solely verbal. For if in a calculation I arrive at this series nowrap|1=1 − 1 + 1 − 1 + 1 − 1 etc. and if in its place I substitute 1/2, no one will rightly impute to me an error, which however everyone would do had I put some other number in the place of this series. Whence no doubt can remain that in fact the series nowrap|1=1 − 1 + 1 − 1 + 1 − 1 + etc. and the fraction 1/2 are equivalent quantities and that it is always permitted to substitute one for the other without error. Thus the whole question is seen to reduce to this, whether we call the fraction 1/2 the correct sum of nowrap|1=1 − 1 + 1 − 1 + etc.; and it is strongly to be feared that those who insist on denying this and who at the same time do not dare to deny the equivalence have stumbled into a battle over words.

But I think all this wrangling can be easily ended if we should carefully attend to what follows… [Euler 1760 §10 and beginning of §11, p.211; English translation by Barbeau and Leah (p.148)]

Euler also used finite differences to attack nowrap|1=1 − 1 + 1 − 1 + · · ·. In modern terminology, he took the Euler transform of the sequence and found that it equalled ¹⁄₂. [Grattan-Guinness pp.68-69] As late as 1864, De Morgan claims that "this transformation has always appeared one of the strongest presumptions in favour of nowrap|1=1 − 1 + 1 − … being ¹⁄₂." [De Morgan p.10]

Dilution and new values

Despite the confident tone of his papers, Euler expressed doubt over divergent series in his correspondence with Nicolaus I Bernoulli. Euler claimed that his attempted definition had never failed him, but Bernoulli pointed out a clear weakness: it does not specify how one should determine "the" finite expression that generates a given infinite series. Not only is this a practical difficulty, it would be theoretically fatal if a series were generated by expanding two expressions with different values. Euler's treatment of nowrap|1=1 − 1 + 1 − 1 + · · · rests upon his firm belief that ¹⁄₂ is the only possible value of the series; what if there were another?

In a 1745 letter to Christian Goldbach, Euler claimed that he was not aware of any such counterexample, and in any case Bernoulli had not provided one. Several decades later, when Jean-Charles Callet finally asserted a counterexample, it was aimed at nowrap|1=1 − 1 + 1 − 1 + · · ·. The background of the new idea begins with Daniel Bernoulli in 1771. [Hardy p.14; Bromwich p.322]

Daniel Bernoulli

*cite journal |last=Bernoulli |first=Daniel |title=De summationibus serierum quarunduam incongrue veris earumque interpretatione atque usu |journal= Novi Commentarii academiae scientiarum imperialis Petropolitanae |volume=16 |year=1771 |pages=71–90

Daniel Bernoulli, who accepted the probabilistic argument that nowrap|1=1 − 1 + 1 − 1 + · · · = ¹⁄₂, noticed that by inserting 0s into the series in the right places, it could achieve any value between 0 and 1. In particular, the argument suggested that:1 + 0 − 1 + 1 + 0 − 1 + 1 + 0 − 1 + · · · = ²⁄₃. [Sandifer p.1]

Callet and Lagrange

In a memorandum sent to Joseph Louis Lagrange toward the end of the century, Callet pointed out that nowrap|1=1 − 1 + 1 − 1 + · · · could also be obtained from the series:$frac\{1+x\}\{1+x+x^2\}=1-x^2+x^3-x^5+x^6-x^8+cdots;$substituting "x" = 1 now suggests a value of ²⁄₃, not ¹⁄₂. Lagrange approved Callet's submission for publication in the "Mémoires" of the French Academy of Sciences, but it was never directly published. Instead, Lagrange (along with Charles Bossut) summarized Callet's work and responded to it in the "Mémoires" of 1799. He defended Euler by suggesting that Callet's series actually should be written with the 0 terms left in::$1+0-x^2+x^3+0-x^5+x^6+0-x^8+cdots,$which reduces to:1 + 0 − 1 + 1 + 0 − 1 + 1 + 0 − 1 + · · ·instead. [Bromwich pp.319-320, Lehmann p.176, Kline 1972 p.463; here Bromwich seems to cite Borel's "Leçons sur les Séries Divergentes", pp.1-10.]

19th century

The 19th century is remembered as the approximate period of Cauchy's and Abel's largely successful ban on the use of divergent series, but Grandi's series continued to make occasional appearances. Some mathematicians did not follow Abel's lead, mostly outside of France, and British mathematicians especially took "a long time" to understand the analysis coming from the continent. [Hardy p.18]

In 1803, Robert Woodhouse proposed that nowrap|1=1 − 1 + 1 − 1 + · · · summed to something called:$frac\{1\}\{1+1\},$which could be distinguished from ¹⁄₂. Ivor Grattan-Guinness remarks on this proposal, "… R. Woodhouse … wrote with admirable honesty on the problems which he failed to understand. … Of course, there is no harm in defining new symbols such as ¹⁄₁₊₁; but the idea is 'formalist' in the unflattering sense, and it does not bear on the problem of the convergence of series." [Grattan-Guinness p.71]

Algebraic reasoning

In 1830, a mathematician identified only as "M. R. S." wrote in the "Annales de Gergonne" on a technique to numerically find fixed points of functions of one variable. If one can transform a problem into the form of an equation "x = A + f(x)", where "A" can be chosen at will, then :$x\; =\; A+f(A+f(A+f(cdots)))$should be a solution, and truncating this infinite expression results in a sequence of approximations. Conversely, given the series nowrap|1="x" = "a" − "a" + "a" − "a" + · · ·, the author recovers the equation:$x\; =\; a\; -\; x,$to which the solution is (¹⁄₂)"a".

M. R. S. notes that the approximations in this case are "a", 0, "a", 0, …, but there is no need for Leibnitz's "subtle reasoning". Moreover, the argument for averaging the approximations is problematic in a wider context. For equations not of the form "x = A + f(x)", M. R. S.'s solutions are continued fractions, continued radicals, and other infinite expressions. In particular, the expression nowrap|1="a" / ("a" / ("a" / · · · ))) should be a solution of the equation nowrap|1="x" = "a"/"x". Here, M. R. S. writes that based on Leibnitz's reasoning, one is tempted to conclude that "x" is the average of the truncations "a", 1, "a", 1, …. This average is nowrap|1=(1 + "a")/2, but the solution to the equation is the square root of "a". [M.R.S. pp.363-365]

Bernard Bolzano criticized M. R. S.' algebraic solution of the series. In reference to the step:$x\; =\; a-a+a-a+cdots\; =\; a-(a-a+a-cdots),$Bolzano charged,This comment exemplifies Bolzano's intuitively appealing but deeply problematic views on infinity. In his defense, Cantor himself pointed out that Bolzano worked in a time when the concept of the cardinality of a set was absent. [Sbaragli p.27; the primary source for Bolzano is not given, but it appears to be Moreno and Waldegg (1991), "The conceptual evolution of actual mathematical infinity". "Educational studies in mathematics". 22, 211-231. The primary source for Cantor is his 1932 "Gesammelte Abhandlngen".]

De Morgan and company

As late as 1844, Augustus De Morgan commented that if a single instance where nowrap|1=1 − 1 + 1 − 1 + · · · did not equal ¹⁄₂ could be given, he would be willing to reject the entire theory of trigonometric series. [Kline 1972 p.976]

The same volume contains papers by Samuel Earnshaw and J R Young dealing in part with nowrap|1=1 − 1 + 1 − 1 + · · ·. G. H. Hardy dismisses both of these as "little more than nonsense", in contrast to De Morgan's "remarkable mixture of acuteness and confusion"; in any case, Earnshaw got De Morgan's attention with the following remarks:

De Morgan fired back in 1864 in the same journal:

Frobenius and modern mathematics

The last scholarly article to be motivated by 1 − 1 + 1 − 1 + · · · might be identified as the first article in the modern history of divergent series. [For example, it is presented as such in Smail pp.3-4.] Georg Frobenius published an article titled "Ueber die Leibnitzsche Reihe" ("On Leibniz's series") in 1880. He had found Leibniz's old letter to Wolff, citing it along with a 1836 article by J. L. Raabe, who in turn drew on ideas by Leibniz and Daniel Bernoulli. [Raabe p.355; Frobenius p.262]

Frobenius' short paper, barely two pages, begins by quoting from Leibniz's treatment of 1 − 1 + 1 − 1 + · · ·. He infers that Leibniz was actually stating a generalization of Abel's Theorem. The result, now known as Frobenius' theorem, [Smail p.4] has a simple statement in modern terms: any series that is Cesàro summable is also Abel summable to the same sum. Historian Giovanni Ferraro emphasizes that Frobenius did not actually state the theorem in such terms, and Leibniz did not state it at all. Leibniz was defending the association of the divergent series nowrap|1 − 1 + 1 − 1 + · · · with the value ¹⁄₂, while Frobenius' theorem is stated in terms of convergent sequences and the epsilon-delta formulation of the limit of a function. [Ferraro 1999 p.116]

Frobenius' theorem was soon followed with further generalizations by Otto Hölder and Thomas Joannes Stieltjes in 1882. Again, to a modern reader their work strongly suggests new definitions of the sum of a divergent series, but those authors did not yet make that step. Ernesto Cesàro proposed a systematic definition for the first time in 1890. [Ferraro 1999 pp.117, 128] Since then, mathematicians have explored many different summability methods for divergent series. Most of these, especially the simpler ones with historical parallels, sum Grandi's series to ¹⁄₂. Others, motivated by Daniel Bernoulli's work, sum the series to another value, and a few do not sum it at all.

Notes

References

;Cited primary sourcesThe full texts of many of the following references are publically available on the Internet from Google Books; the Euler archive at Dartmouth College; DigiZeitschriften, a service of Deutsche Forschungsgemeinschaft; or Gallica, a service of the Bibliothèque nationale de France.

*cite book |last=Bougainville |first=Louis Antoine de |authorlink=Louis Antoine de Bougainville |title=Traité du calcul intégral |year=1754 |publisher=Guérin and Delatour |url=http://books.google.com/books?vid=OCLC32646505
*cite journal |last=De Morgan |first=Augustus |authorlink=Augustus De Morgan |date=1844-03-04 |title=On Divergent Series, and various Points of Analysis connected with them |journal=Transactions of the Cambridge Philosophical Society |volume=8 |issue=2 |pages=182–203
*cite journal |author=       |date=1864-05-16 |title=A Theorem relating to Neutral Series |journal=Transactions of the Cambridge Philosophical Society |volume=11 |issue=1 |url=http://books.google.com/books?vid=0C7aIzhHUS1NN6WD&id=UTQAAAAAQAAJ
*cite journal |last=Earnshaw |first=Samuel |authorlink=Samuel Earnshaw |title=On the Values of the Sine and Cosine of an Infinite Angle |journal=Transactions of the Cambridge Philosophical Society |volume=8 |issue=3 |date=1844-12-09 |pages=255–268 Written November 9, 1844.
*cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |title=Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum |year=1755 |url=http://www.math.dartmouth.edu/~euler/pages/E212.html
*cite journal |author=       |title=De seriebus divergentibus |journal=Novi Commentarii academiae scientiarum Petropolitanae |volume=5 |year=1760 |pages=205–237 |url=http://www.math.dartmouth.edu/~euler/pages/E247.html
*cite journal |last=Frobenius |first=Ferdinand Georg |authorlink=Ferdinand Georg Frobenius |title=Ueber die Leibnitzsche Reihe |journal=Journal für die reine und angewandte Mathematik |volume=89 |year=1880 |pages=262–264 |url=http://www.digizeitschriften.de/resolveppn/PPN243919689_0089 Written in October 1879.
*cite journal |last=Lagrange |first=Joseph Louis |authorlink=Joseph Louis Lagrange |title=Rapport sur un mémoire présenté à la classe par le citoyen Callet |journal=Mémoires de l'Institut des sciences, lettres et arts. Sciences mathématiques et physiques |year=1796, pub.1799 |volume=3 |pages=1–11 |url=http://gallica.bnf.fr/ark:/12148/bpt6k3218t/f8.table
*cite book |last=Leibniz |first=Gottfried |authorlink=Gottfried Leibniz |year=2003 |title=Sämtliche Schriften und Briefe, Reihe 7, Band 3: 1672–1676: Differenzen, Folgen, Reihen |S. Probst, E. Knobloch, N. Gädeke, editors |publisher=Akademie Verlag |id=ISBN 3-05-004003-3 |url=http://www.nlb-hannover.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/
*cite journal |author=M.R.S. |title=Analyse algébrique. Note sur quelques expressions algébriques peu connues |journal=Annales de Gergonne |volume=20 |year=1830 |pages=352–366 |url=http://www.numdam.org/item?id=AMPA_1829-1830__20__352_1
*cite journal |last=Raabe |first=J. L. |authorlink=J. L. Raabe |title=Über die Summation periodischer Reihen und die Reduction des Integrals |journal=Journal für die reine und angewandte Mathematik |volume=15 |year=1836 |pages=355–364 |url=http://www.digizeitschriften.de/resolveppn/GDZPPN002140764
*cite journal |last=Varignon |first=Pierre |authorlink=Pierre Varignon |title=Précautions à prendre dans l'usage des Suites ou Series infinies résultantes, tant de la division infinie des fractions, que du Développement à l'infini des puissances d'exposants négatifs entiers |year=1715 |journal=Histoire de l'Academie royale des sciences |pages=203–225 |url=http://gallica.bnf.fr/ark:/12148/bpt6k3592w/f343.table
*cite journal |last=Young |first=John Radford |authorlink=John Radford Young |date=1846-12-07 |title=On the Principle of Continuity, in reference to certain Results of Analysis |journal=Transactions of the Cambridge Philosophical Society |volume=8 |issue=4 |pages=429–440

;Cited secondary sources
*cite journal |last=Bagni |first=Giorgio T. |title=Infinite Series from History to Mathematics Education |journal=International Journal for Mathematics Teaching and Learning |date=2005-06-30 |url=http://www.cimt.plymouth.ac.uk/journal/bagni.pdf
*cite web |author=       |title=Appunti di Storia per la Didattica della Matematica |url=http://www.syllogismos.it/history/appuntistoria.htm
*cite journal |author=Barbeau, E.J., and P.J. Leah |title=Euler's 1760 paper on divergent series |year=1976 |month=May |journal=Historia Mathematica |volume=3 |issue=2 |pages=141–160 |doi=10.1016/0315-0860(76)90030-6
*cite book |last=Bromwich |first=T.J. |year=1926 |origyear=1908 |edition=2e |title=An Introduction to the Theory of Infinite Series
*cite book |last=Cantor |first=Moritz |title=Vorlesungen über Geschichte der Mathematik: Dritter Band Vom Jahre 1668 bis zum Jahre 1758 |origyear=1901 |year=1965 |edition=2e |location=New York |publisher=Johnson Reprint Corporation |id=LCC|QA21|.C232|1965
*cite journal |last=Dutka |first=Jacques |title=On the St. Petersburg Paradox |journal=Archive for History of Exact Sciences |year=1988 |month=March |volume=39 |issue=1 |pages=13–39 |doi=10.1007/BF00329984
*cite journal |last=Ferraro |first=Giovanni |title=The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics |journal=Archive for History of Exact Sciences |year=1999 |month=June |volume=54 |issue=2 |pages=101–135 |doi=10.1007/s004070050036
*cite journal |author=       |title=Analytical Symbols and Geometrical Figures in Eighteenth-Century Calculus |journal=Studies in History and Philosophy of Science |volume=32 |issue=3 |year=2000 |pages=535–555 |doi=10.1016/S0039-3681(01)00009-7
*cite journal |author=       |title=Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730 |journal=Annals of Science |volume=59 |year=2002 |pages=179–199 |doi=10.1080/00033790010028179
*cite journal |author=       |title=Convergence and formal manipulation in the theory of series from 1730 to 1815 |journal=Historia Mathematica |year=2005 |doi=10.1016/j.hm.2005.08.004 |volume=34 |pages=62
*cite journal |author=       and Marco Panza |title=Developing into series and returning from series: A note on the foundations of eighteenth-century analysis |journal=Historia Mathematica |volume=30 |issue=1 |year=2003 |month=February |pages=17–46 |doi=10.1016/S0315-0860(02)00017-4
*cite book |last=Gerhardt |first=C.I. |title=Briefwechsel zwischen Leibniz und Christian Wolf aus den handschriften der Koeniglichen Bibliothek zu Hannover |year=1860 |location=Halle |publisher=H.W. Schmidt |url=http://books.google.com/books?id=5ScCAAAAQAAJ&pg=RA1-PP14
*cite book |last=Grattan-Guinness |authorlink=Ivor Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0
*cite book |last=Hardy |first=G.H. |authorlink=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id=LCC|QA295|.H29|1967
*cite conference |last=Hitt |first=Fernando |title=L’argumentation, la preuve et la démonstration dans la construction des mathématiques : des entités conflictuelles ? une lettre de Godefroy Guillaume Leibnitz à Chrétien Wolf (1713) |booktitle=GDM 2005: Raisonnement mathématique et formation citoyenne |year=2005 |url=http://math.uqam.ca/Actes-GDM-05.pdf
*cite book |last=Kline |first=Morris |authorlink=Morris Kline |title=Mathematical thought from ancient to modern times |origyear=1972 |year=1990 |edition=3-volume ppk ed. |publisher=Oxford UP |id=ISBN 0-19-506136-5
*cite journal |author=       |title=Euler and Infinite Series |journal=Mathematics Magazine |volume=56 |issue=5 |year=1983 |month=November |pages=307–314 |url=http://links.jstor.org/sici?sici=0025-570X%28198311%2956%3A5%3C307%3AEAIS%3E2.0.CO%3B2-M
*cite journal |last=Knobloch |first=Eberhard |title=Beyond Cartesian limits: Leibniz’s passage from algebraic to “transcendental” mathematics |journal=Historia Mathematica |volume=33 |year=2006 |pages=113–131 |doi=10.1016/j.hm.2004.02.001
*cite book |last=Knopp |first=Konrad |authorlink=Konrad Knopp |title=Theory and Application of Infinite Series |year=1990 |origyear=1922 |publisher=Dover |id=ISBN 0-486-66165-2
*cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1
*cite book |last=Markushevich |first=A.I. |title=Series: fundamental concepts with historical exposition |year=1967 |edition=English translation of 3rd revised edition (1961) in Russian |publisher=Hindustan Pub. Corp. |id=LCC|QA295|.M333|1967
*cite book |author=Mazzone, Silvia and Clara Silvia Roero |title=Jacob Hermann and the diffusion of the Leibnizian calculus in Italy |publisher=Leo S. Olschki |year=1997 |id=ISBN 88-222-4555-5
*cite book |last=Moore |first=Charles |title=Summable Series and Convergence Factors |publisher=AMS |year=1938 |id=LCC|QA1|.A5225|V.22
*cite journal |last=Panza |first=Marco |year=1992 |journal=Cahiers d'Histoire de Philosophie des Sciences |title=La forma della quantità |volume=38 Volume 1, Parte III, Cap. 1, "La questione della serie di Grandi (1696 - 1715)", pp.296-345.
*cite book |last=Reiff |first=Richard |title=Geschichte der unendlichen Reihen |origyear=1889 |year=1969 |publisher=Martin Sändig oHG |id=LCC|QA295|.R39|1969 Reiff's German-language work "History of infinite Series" is frequently cited by other sources when they deal with the history of Grandi's series. Hardy (p.21) calls it "useful but uninspiring and not always accurate."
*cite web |last=Sandifer |first=Ed |year=2006 |month=June |title=Divergent series |work=How Euler Did It |publisher=MAA Online |url=http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%20series.pdf
*cite web |last=Sbaragli |first=Silvia |year=2004 |title=Teacher's convictions on mathematical infinity |url=http://math.unipa.it/~grim/thesis_sbaragli__04_cap1_engl.pdf
*cite book |last=Schubring |first=Gert |title=Conflicts between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17-19th Century France and Germany |year=2005 |publisher=Springer |id=ISBN 0-387-22836-5, LCC|QA300|.S377|2005
*cite book |last=Smail |first=Lloyd |title=History and Synopsis of the Theory of Summable Infinite Processes |year=1925 |publisher=University of Oregon Press |id=LCC|QA295|.S64
* D. J. Struik, editor, "A source book in mathematics, 1200-1800" (Princeton University Press, Princeton, New Jersey, 1986). ISBN 0-691-08404-1, ISBN 0-691-02397-2 (pbk). See in particular pp. 178-180 in regard to the "versiera" (i.e. witch of Agnesi) and Maria Gaetana Agnesi (1718-1799) of Milan, sister of the composer Maria Teresa Agnesi, the first important woman mathematician since Hypatia (fifth century A.D.).
*cite book |author=Upham, Thomas Cogswell and Dugald Stewart |year=1831 |title=Elements of Mental Philosophy |publisher=Hillard, Gray & Co. |url=http://books.google.com/books?vid=OCLC03633435
*cite book |author=Weidlich |first=John E. |title=Summability methods for divergent series |year=1950 |month=June |publisher=Stanford M.S. theses

;Further reading
*cite book |last=Eves |first=Howard W. |authorlink=Howard Eves |title=In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes |year=2002 |publisher=The Mathematical Association of America |id=ISBN 0-88385-542-9