History of Grandi's series

Main article: 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 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:

"By putting parentheses into the expression 1 − 1 + 1 − 1 + · · · in different ways, I can, if I want, obtain 0 or 1. But then the idea of the creation ex nihilo is perfectly plausible."

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.

(1, ¹⁄₂) on the witch of Agnesi

Grandi's mathematical treatment of 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 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 = a³/(a² + x²) has the series expansion

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{a^{2n-1}}}=a-{\frac {x^{2}}{a}}+{\frac {x^{4}}{a^{3}}}-{\frac {x^{6}}{a^{5}}}+\cdots$

and setting a = x = 1, one has 1 − 1 + 1 − 1 + · · · = ¹⁄₂.

• According to 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 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."

Grandi offered a new explanation that 1 − 1 + 1 − 1 + · · · = ¹⁄₂ in 1710, both in the second edition of the Quadratura circula and in a new work, De Infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica. 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.

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:

"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."

Marchetti

(Further on Marchetti...)

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. 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.

Background

As early as 1674, in a minor, lesser-known writing De Triangulo Harmonico on the harmonic triangle, Leibniz mentioned 1 − 1 + 1 − 1 + · · · very briefly in an example:

${\frac {1}{1+1}}={\frac {1}{1}}-{\frac {1}{1+1}}.\;\mathrm {Ergo} \;{\frac {1}{1+1}}=1-1+1-1+1-1\;\mathrm {etc.}$

The series 1 − 1 + 1 − 1 + · · · also appears indirectly in a discussion with Tschirnhaus in 1676.

Leibniz had already considered the divergent alternating series 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 a finite: "Now normally nature chooses the middle if neither of the two is permitted, or rather if it cannot be determined which of the two is permitted, and the whole is equal to a finite quantity." Leibniz did not assert that the series had a sum, but he did claim it was equal to ¹⁄₃ following Mercator's method. The attitude that a series could equal some finite quantity without actually adding up to it as a sum would be commonplace in the 18th century, although no distinction is made in modern mathematics.

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.

Solutions

Beginning of the published Leibniz-Wolff letter

In the 1710s, Leibniz described Grandi's series in his correspondence with several other mathematicians. The letter with the most lasting impact was his first reply to Wolff, which he published in the Acta Eruditorum.

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 1 − 1 + 1 − 1 + · · · = ¹⁄₂, claiming that the relation was well-founded because there existed a "demonstratio linearis" — a geometric demonstration.

On the other hand, Leibniz sharply criticized Grandi's example of the shared gem, claiming that the series 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 thought that the argument from 1/(1 + x) was valid; he took it as an example of his law of continuity. Still, he thought that one should be able to find the sum of the series 1 − 1 + 1 − 1 + · · · directly, without needing to refer back to the expression 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. 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. Yet Leibniz invented an argument for Grandi's series that came surprisingly close to a modern method.

For Grandi's series, Leibniz fell back upon more metaphysical reasoning. Since the partial sums of the series are 0 or 1 with equal "probability", one should take the true value or sum of the series to be their mean, which is 1/2. Eli Maor says of this solution, "Such a brazen, careless reasoning indeed seems incredible to us today…" 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."

(Leibniz to Pierre Dangicourt (1716)).

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 was so pleased with the solution that he sought to extend the arithmetic mean method to more divergent series such as 1 − 2 + 4 − 8 + 16 − · · ·. Briefly, if one expresses a partial sum of this series as a function of the penultimate term, one obtains either (4m + 1)/3 or (−4n + 1)/3. The mean of these values is (2m −2n + 1)/3, and assuming that m = n at infinity yields 1/3 as the value of the series. 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. The arithmetic means of neighboring partial sums do not converge to any particular value, and for all finite cases one has n = 2m, not n = m. Generally, the terms of a summable series should decrease to zero; even 1 − 1 + 1 − 1 + · · · could be expressed as a limit of such series. Leibniz counsels Wolff to reconsider so that he "might produce something worthy of science and himself."

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.

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."

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 dividing polynomials (essentially long division) to the ratio k/(m + n), he noticed that one always had a remainder. If 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".

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. 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 from memory. Upon returning to Paris and checking in at the Academy, Varignon soon discovered that the great Leibniz had ruled in favor of Grandi. Varignon still had to revise his paper by looking up and including the citation to Jacob Bernoulli. Rather than 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.

(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.)

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. 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 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 1 − 1 = n − n is guaranteed by 1 + n = n + 1. He concludes that the fundamental mistake is in using a divergent series to begin with:

"In fact, it doesn't happen that if we stop this series, the following terms can be neglected in comparison with preceding terms; this property is verified only for convergent series."

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 1/(1 + 1) = 1/2 and yet (1 − 1) + (1 − 1) + · · · = 0.

Euler

Further information: [[:Euler on infinite series]]

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. He identifies the series as being first considered by Leibniz, and he reviews Leibniz's 1713 argument based on the series 1 − a + a² − a³ + a⁴ − a⁵ + · · ·, calling it "fairly sound reasoning", and he also mentions the even/odd median argument. Euler writes that the usual objection to the use of 1/(1 + a) is that it does not equal 1 − a + a² − a³ + a⁴ − a⁵ + · · · 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. He then claims that since an infinite series has no last term, there is no place for the remainder and it should be neglected. After reviewing more badly divergent series like 1 + 2 + 4 + 8 + · · ·, where he judges his opponents to have firmer support, Euler seeks to define away the issue:

"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 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 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 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 also used finite differences to attack 1 − 1 + 1 − 1 + · · ·. In modern terminology, he took the Euler transform of the sequence and found that it equalled ¹⁄₂. As late as 1864, De Morgan claims that "this transformation has always appeared one of the strongest presumptions in favour of 1 − 1 + 1 − … being ¹⁄₂."

Dilution and new values

Despite the confident tone of his papers, Euler expressed doubt over divergent series in his correspondence with N. 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 1 − 1 + 1 − 1 + · · · rests upon his firm assertion 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 invented a counterexample, it was aimed at 1 − 1 + 1 − 1 + · · ·. The background of the new idea begins with Daniel Bernoulli in 1771.

Daniel Bernoulli

• Bernoulli, Daniel (1771). "De summationibus serierum quarunduam incongrue veris earumque interpretatione atque usu". Novi Commentarii academiae scientiarum imperialis Petropolitanae. 16: 71–90.

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. In particular, the argument suggested that

1 + 0 − 1 + 1 + 0 − 1 + 1 + 0 − 1 + · · · = ²⁄₃.

Callet and Lagrange

In a memorandum sent to Joseph Louis Lagrange toward the end of the century, Callet pointed out that 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.

19th century

In 1803, Robert Woodhouse proposed that 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."

In 1830, a mathematician identified only as "M. R. S." wrote in the Annales de Gergonne on algebraic techniques to handle divergent series as well as continued fractions and continued radicals. This paper acknowledge's Leibniz's averaging approach but argues that, for example, the expression a / (a / (a / · · · ))) does not represent the average of the truncated expressions, (1 + a)/2, but rather the solution to the equation x = a/x, which is the square root of a. When the author encounters x = a − a + a − a + · · ·, it is handled similarly, as the root of the equation x = a - x, which is x = a/2. Bernard Bolzano criticized this solution, saying of the written-out form

$x=a-(a-a+a-a+a-a+\cdots )=a-x$

that "The series within parenthses has clearly not the same set of numbers of that originally indicated with x, as the first term a is missing."

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.

"I do not argue with those who reject everything that is not within the providence of arithmetic, but only with those who abandon the use of infinitely divergent series and yet appear to employ finitely divergent series with confidence. Such appears to be the practice, both at home and abroad. They seem perfectly reconciled to 1 − 1 + 1 − 1 + · · ·, but cannot admit 1 + 2 + 4 + · · · = −1."

"The whole fabric of periodic series and integrals … would fall instantly if it were shown to be possible that 1 − 1 + 1 − 1 + · · · might be one quantity as a limiting form of A₀ − A₁ + A₂ − · · ·and another as a limiting form of A₀ − A₁ + A₂ − · · ·."

Earnshaw:

"It is not very unusual to cast a mantle of mystery over this subject, by introducing zeros… But such a device, however much it may serve to satisfy the eye, cannot satisfy the head…"

de Morgan:

"I cannot approve of introducing ciphers to satisfy the eye: but to me they always introduced themselves."

"…those who reject casual evanescents out of a routine of operation have no right to charge those who do not reject with introduction."

Frobenius and Cesaro

Main article: Summation of Grandi's series

References

Cited primary sources

• Bougainville, Louis Antoine de (1754). Traité du calcul intégral. Guérin and Delatour.
• De Morgan, Augustus (1844-03-04). "On Divergent Series, and various Points of Analysis connected with them". Transactions of the Cambridge Philosophical Society. 8 (2): 182–203.
•        (1864-05-16). "A Theorem relating to Neutral Series". Transactions of the Cambridge Philosophical Society. 11 (1).{{cite journal}}: CS1 maint: multiple names: authors list (link) The full text is publically available on the Internet from Google Books.
• Earnshaw, Samuel (1844-12-09). "On the Values of the Sine and Cosine of an Infinite Angle". Transactions of the Cambridge Philosophical Society. 8 (3): 255–268. Written November 9, 1844.
• Euler, Leonhard (1755). Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum.
•        (1760). "De seriebus divergentibus". Novi Commentarii academiae scientiarum Petropolitanae. 5: 205–237.{{cite journal}}: CS1 maint: multiple names: authors list (link) The full texts are publically available on the Internet from the Euler archive.
• Frobenius, Ferdinand Georg (1880). "Über die Leibnitzsche Reihe". Journal für die reine und angewandte Mathematik. 89: 262–264. The full text is available from DigiZeitschriften, a service of Deutsche Forschungsgemeinschaft.
• Lagrange, Joseph Louis (1796, pub.1799). "Rapport sur un mémoire présenté à la classe par le citoyen Callet". Mémoires de l'Institut des sciences, lettres et arts. Sciences mathématiques et physiques. 3: 1–11. {{cite journal}}: Check date values in: |year= (help) The full text is publically available on the Internet from the Bibliothèque nationale de France.
• Leibniz, Gottfried (2003). Sämtliche Schriften und Briefe, Reihe 7, Band 3: 1672–1676: Differenzen, Folgen, Reihen. Akademie Verlag. ISBN 3-05-004003-3. {{cite book}}: Text "S. Probst, E. Knobloch, N. Gädeke, editors" ignored (help)
• M.R.S. (1830). "Analyse algébrique. Note sur quelques expressions algébriques peu connues". Annales de Gergonne. 20: 352–366.
• Varignon, Pierre (1715). "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". Histoire de l'Academie royale des sciences: 203–225. The full text is publically available on the Internet from the Bibliothèque nationale de France.
• Young, John Radford (1846-12-07). "On the Principle of Continuity, in reference to certain Results of Analysis". Transactions of the Cambridge Philosophical Society. 8 (4): 429–440.

Cited secondary sources

• Bagni, Giorgio T. (2005-06-30). "Infinite Series from History to Mathematics Education" (PDF). International Journal for Mathematics Teaching and Learning.
•       . "Appunti di Storia per la Didattica della Matematica".{{cite web}}: CS1 maint: multiple names: authors list (link)
• Barbeau, E.J., and P.J. Leah (1976). "Euler's 1760 paper on divergent series". Historia Mathematica. 3 (2): 141–160. doi:10.1016/0315-0860(76)90030-6. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: multiple names: authors list (link)
• Bromwich, T.J. (1926) [1908]. An Introduction to the Theory of Infinite Series (2e ed.).
• Cantor, Moritz (1965) [1901]. Vorlesungen über Geschichte der Mathematik: Dritter Band Vom Jahre 1668 bis zum Jahre 1758 (2e ed.). New York: Johnson Reprint Corporation. LCC QA21 .C232 1965.
• Dutka, Jacques (1988). "On the St. Petersburg Paradox". Archive for History of Exact Sciences. 39 (1): 13–39. doi:10.1007/BF00329984. {{cite journal}}: Unknown parameter |month= ignored (help)
• Ferraro, Giovanni (2000). "Analytical Symbols and Geometrical Figures in Eighteenth-Century Calculus". Studies in History and Philosophy of Science. 32 (3): 535–555. doi:10.1016/S0039-3681(01)00009-7.
•        (2002). "Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730". Annals of Science. 59: 179–199. doi:10.1080/00033790010028179.{{cite journal}}: CS1 maint: multiple names: authors list (link)
•        (2005). "Convergence and formal manipulation in the theory of series from 1730 to 1815". Historia Mathematica. doi:10.1016/j.hm.2005.08.004.{{cite journal}}: CS1 maint: multiple names: authors list (link)
•        and Marco Panza (2003). "Developing into series and returning from series: A note on the foundations of eighteenth-century analysis". Historia Mathematica. 30 (1): 17–46. doi:10.1016/S0315-0860(02)00017-4. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: multiple names: authors list (link)
• Gerhardt, C.I. (1860). Briefwechsel zwischen Leibniz und Christian Wolf aus den handschriften der Koeniglichen Bibliothek zu Hannover. Halle: H.W. Schmidt.
• Grattan-Guinness, Ivor (1970). The development of the foundations of mathematical analysis from Euler to Riemann. MIT Press. ISBN 0-262-07034-0.
• Hardy, G.H. (1949). Divergent Series. Clarendon Press. LCC QA295 .H29 1967.
• Hitt, Fernando (2005). "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)" (PDF). GDM 2005: Raisonnement mathématique et formation citoyenne. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help)
• Kline, Morris (1990) [1972]. Mathematical thought from ancient to modern times (3-volume ppk ed. ed.). Oxford UP. ISBN 0-19-506136-5. {{cite book}}: |edition= has extra text (help)
•        (1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: multiple names: authors list (link)
• Knobloch, Eberhard (2006). "Beyond Cartesian limits: Leibniz's passage from algebraic to "transcendental" mathematics". Historia Mathematica. 33: 113–131. .
• Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. ISBN 0-486-66165-2.
• Maor, Eli (1987). To infinity and beyond: a cultural history of the infinite. Birkhäuser. ISBN 3-7643-3325-1.
• Markushevich, A.I. (1967). Series: fundamental concepts with historical exposition (English translation of 3rd revised edition (1961) in Russian ed.). Hindustan Pub. Corp. LCC QA295 .M333 1967.
• Mazzone, Silvia and Clara Silvia Roero (1997). Jacob Hermann and the diffusion of the Leibnizian calculus in Italy. Leo S. Olschki. ISBN 88-222-4555-5.
• Moore, Charles (1938). Summable Series and Convergence Factors. AMS. LCC QA1 .A5225 V.22.
• Panza, Marco (1992). "La forma della quantità". Cahiers d'Histoire de Philosophie des Sciences. 38. Volume 1, Parte III, Cap. 1, "La questione della serie di Grandi (1696 - 1715)", pp.296-345.
• Reiff, Richard (1969) [1889]. Geschichte der unendlichen Reihen. Martin Sändig oHG. 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."
• Sandifer, Ed (2006). "Divergent series" (PDF). How Euler Did It. MAA Online. {{cite web}}: Unknown parameter |month= ignored (help)
• Sbaragli, Silvia (2004). "Teacher's convictions on mathematical infinity" (PDF).
• Schubring, Gert (2005). Conflicts between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17-19th Century France and Germany. Springer. ISBN 0-387-22836-5, LCC QA300 .S377 2005.
• Smail, Lloyd (1925). History and Synopsis of the Theory of Summable Infinite Processes. University of Oregon Press. 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.).
• Upham, Thomas Cogswell and Dugald Stewart (1831). Elements of Mental Philosophy. Hillard, Gray & Co.
• Weidlich, John E. (1950). Summability methods for divergent series. Stanford M.S. theses. {{cite book}}: Unknown parameter |month= ignored (help)

Further reading

• Eves, Howard W. (2002). In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes. The Mathematical Association of America. ISBN 0-88385-542-9.