In 1887, Cesàro came close to stating the definition of (C, n) summation, but he gave only a few examples. It has been proven that (C, n) summation and (H, n) summation always give the same results, but they have different historical backgrounds. The other commonly formulated generalization of Cesàro summation is the sequence of (C, n) methods. guarantees that it is the Abel sum as well this will also be proved directly below. The fact that 1⁄ 4 is the (H, 2) sum of 1 − 2 + 3 − 4 +. The "H" stands for Otto Hölder, who first proved in 1882 what mathematicians now think of as the connection between Abel summation and (H, n) summation 1 − 2 + 3 − 4 +. Above, the even means converge to 1⁄ 2, while the odd means are all equal to 0, so the means of the means converge to the average of 0 and 1⁄ 2, namely 1⁄ 4. The (H, 1) sum is Cesàro summation, and higher methods repeat the computation of means. There are two well-known generalizations of Cesàro summation: the conceptually simpler of these is the sequence of (H, n) methods for natural numbers n.
This sequence of means does not converge, so 1 − 2 + 3 − 4 +. In the case where a n = b n = (−1) n, the terms of the Cauchy product are given by the finite diagonal sums
The Cauchy product of two infinite series is defined even when both of them are divergent. is the Cauchy product (discrete convolution) of 1 − 1 + 1 − 1 +. The details on his summation method are below the central idea is that 1 − 2 + 3 − 4 +. and asserts that both the sides are equal to 1⁄ 4." For Cesàro, this equation was an application of a theorem he had published the previous year, which is the first theorem in the history of summable divergent series. In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously brought into calculus, pointing out, "One already writes (1 − 1 + 1 − 1 +. Such a method must also sum Grandi's series as 1 − 1 + 1 − 1 +.