History of Grandi's series
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 Grandi offered a new explanation that 1 − 1 + 1 − 1 + · · · = 1⁄2 in 1710, both in the second edition of the Quadratura circula[4] and in a new work, De Infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica.
[5] 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.
The second repeats the link between the series and the creation of the universe by God: Sed inquies: aggregatum ex infinitis differentiis infinitarum ipsi DV æqualium, sive continuè, sive alternè sumptarum, est demum summa ex infinitis nullitatibus, seu 0, quomodo ergo quantitatem notabilem aggreget?
At repono, eam Infiniti vim agnoscendam, ut etiam quod per se nullum est multiplicando, in aliquid commutet, sicuti finitam magnitudiné dividendo, in nullam degenerare cogit: unde per infinitam Dei Creatoris potentiam omnia ex nihlo facta, omniaque in nihilum redigi posse: neque adeò absurdum esse, quantitatem aliquam, ut ita dicam, creari per infinitam vel multiplicationem, vel additionem ipsius nihili, aut quodvis quantum infinita divisione, aut subductione in nihilum redigit.
[8] 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.
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.
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.
In general, Leibniz believed that the algorithms of calculus were a form of "blind reasoning" that ultimately had to be founded upon geometrical interpretations.
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.
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.
[21] 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.
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.
[24] 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.
[25] 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.
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.
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.
)[32] In a 1715 letter to Jacopo Riccati, Leibniz mentioned the question of Grandi's series and advertised his own solution in the Acta Eruditorum.
Louis Antoine de Bougainville briefly treats the series in his acclaimed 1754 textbook Traité du calcul intégral.
[35] Leonhard Euler treats 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.
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 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.
[40] Daniel Bernoulli, who accepted the probabilistic argument that 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.
"[44] 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.
Conversely, given the series x = a − a + a − a + · · ·, the author recovers the equation 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 Leibniz's "subtle reasoning".
[46] As late as 1844, Augustus De Morgan commented that if a single instance where 1 − 1 + 1 − 1 + · · · did not equal 1⁄2 could be given, he would be willing to reject the entire theory of trigonometric series.
Leibniz was defending the association of the divergent series 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.
