# Category:Divergent series

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## Pages in category "Divergent series"

The following 14 pages are in this category, out of 14 total, this list may not reflect recent changes (learn more).

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This category has only the following subcategory.

The following 14 pages are in this category, out of 14 total, this list may not reflect recent changes (learn more).

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1. History of Grandi's series – Guido Grandi reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into 1 −1 +1 −1 + · · · produced varying results, but then the idea of the creation ex nihilo is perfectly plausible. In fact, the series was not a subject for Grandi. Rather, like many mathematicians to follow, he thought the true value of the series was 1⁄2 for a variety of reasons. Grandis mathematical treatment of 1 −1 +1 −1 + · · · occurs in his 1703 book Quadratura circula et hyperbolae per infinitas hyperbolas geometrice exhibita. Broadly interpreting Grandis work, he derived 1 −1 +1 −1 + · · · = 1⁄2 through geometric reasoning connected with his investigation of the witch of Agnesi. 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. Two brothers inherit a priceless gem from their father, whose will forbids them to sell it, if this agreement lasts for all eternity between the brothers descendants, then the two families will each have half possession of the gem, even though it changes hands infinitely often. This argument was 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, 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, Marchetti found the claim that an infinite number of zeros could add up to a finite quantity absurd, and he inferred from Grandis treatment the danger posed by theological reasoning. The two mathematicians began attacking each other in a series of letters, their debate was ended only by Marchettis death in 1714. With the help and encouragement of Antonio Magliabechi, Grandi sent a copy of the 1703 Quadratura to Leibniz, along with a letter expressing compliments, Leibniz received and read this first edition in 1705, and he called it an unoriginal and less-advanced attempt at his calculus. E r g o 11 +1 =1 −1 +1 −1 +1 −1 e t c. Presumably he arrived at this series by repeated substitution,11 +1 =1 −11 +1 =1 −11 +1 =1 − And so on. 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. Two years after that, Leibniz formulated the first convergence test in the history of mathematics, in the 1710s, Leibniz described Grandis series in his correspondence with several other mathematicians. The letter with the most lasting impact was his first reply to Wolff, in this letter, Leibniz attacked the problem from several angles