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 produced varying results: either
or
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 occurs in his 1703 book Quadratura circula et hyperbolae per infinitas hyperbolas geometrice exhibita. Broadly interpreting Grandi's work, he derived 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 = a3/(a2 + x2) has the series expansion
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
and substituted x = 1 to get . 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 in 1710, both in the second edition of the Quadratura circulaPanza (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 very briefly in an example: Leibniz p.369
Presumably he arrived at this series by repeated substitution:
And so on.
The series also appears indirectly in a discussion with Tschirnhaus in 1676.Leibniz p.817
Leibniz had already considered the divergent alternating series 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
Solutions
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 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 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 thought that the argument from was valid; he took it as an example of his law of continuity. Since the relation 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 directly, without needing to refer back to the expression 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 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 justified 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 . 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 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 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 in China, in the hope that Claudio Filippo Grimaldi'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 , he noticed that one always had a remainder.Ferraro 2002 p.181 If then this remainder decreases and "finally is less than any given quantity", and one has
If m = n, then this equation becomes
Bernoulli called this equation a "not inelegant paradox".Knopp p.457Cantor (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…)