History of Grandi's Series - Dilution and New Values

Dilution and New Values

Despite the confident tone of his papers, Euler expressed doubt over divergent series in his correspondence with Nicolaus I 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 belief 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 asserted a counterexample, it was aimed at 1 − 1 + 1 − 1 + · · ·. The background of the new idea begins with Daniel Bernoulli in 1771.