History of Grandi's Series - Frobenius and Modern Mathematics

Frobenius and Modern Mathematics

The last scholarly article to be motivated by 1 − 1 + 1 − 1 + · · · might be identified as the first article in the modern history of divergent series. Georg Frobenius published an article titled "Ueber die Leibnitzsche Reihe" (On Leibniz's series) in 1880. He had found Leibniz's old letter to Wolff, citing it along with a 1836 article by Joseph Ludwig Raabe, who in turn drew on ideas by Leibniz and Daniel Bernoulli.

Frobenius' short paper, barely two pages, begins by quoting from Leibniz's treatment of 1 − 1 + 1 − 1 + · · ·. He infers that Leibniz was actually stating a generalization of Abel's Theorem. The result, now known as Frobenius' theorem, has a simple statement in modern terms: any series that is Cesàro summable is also Abel summable to the same sum. Historian Giovanni Ferraro emphasizes that Frobenius did not actually state the theorem in such terms, and Leibniz did not state it at all. 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.

Frobenius' theorem was soon followed with further generalizations by Otto Hölder and Thomas Joannes Stieltjes in 1882. Again, to a modern reader their work strongly suggests new definitions of the sum of a divergent series, but those authors did not yet make that step. Ernesto Cesàro proposed a systematic definition for the first time in 1890. Since then, mathematicians have explored many different summability methods for divergent series. Most of these, especially the simpler ones with historical parallels, sum Grandi's series to 1⁄₂. Others, motivated by Daniel Bernoulli's work, sum the series to another value, and a few do not sum it at all.