Stage one: the search for a value
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The series is named after the Italian cleric Luigi Guido Grandi (1671–1742), who was a professor at the University of Florence, first of philosophy and then of mathematics. He mentioned it in 1703 in his book Quadratura circula et hyperbolae per infinitas exhibita.
For Grandi, the sum 1 – 1 + 1 – 1 + 1 – 1 + … is a geometric series with common ratio q = −1, so its value can be calculated: it is equal to 1 / (1 – q), or 1 / 2. Several mathematicians justified this value in various ways.
In his book Acta eruditorum (1684), Leibniz observed that formally multiplying (1 – x + x2 – x3 + x4…) by (1 + x) gives 1, since the other terms "cancel each other out in pairs"; hence (1 – x + x2 – x3 + x4…) = 1 / (1 + x). The desired result is therefore obtained by setting x equal to 1.
Leibniz also recovered this value by grouping the terms of the series in pairs:
