History of Grandi's Series

History Of Grandi's Series

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

or

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.

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

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

• According to Morris Kline, Grandi started with the binomial expansion

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.