#PREV 1 2 3 4 5 NEXT SERIES#
The eta function in particular is easier to deal with by Euler's methods because its Dirichlet series is Abel summable everywhere the zeta function's Dirichlet series is much harder to sum where it diverges. Euler had already become famous for finding the values of these functions at positive even integers (including the Basel problem), and he was attempting to find the values at the positive odd integers (including Apéry's constant) as well, a problem that remains elusive today. was the functional equation of the eta function, which leads directly to the functional equation of the Riemann zeta function. Part of Euler's motivation for studying series related to 1 − 2 + 3 − 4 +.
The series are also studied for non-integer values of n these make up the Dirichlet eta function. Finally, in his 1890 Sur la multiplication des séries, Cesàro took a modern approach starting from definitions. as "absurd equalities", and in 1883 Cesàro expressed a typical view of the time that the formulas were false but still somehow formally useful. Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for 1 − 2 n + 3 n − 4 n +. Ĭesàro's teacher, Eugène Charles Catalan, also disparaged divergent series. There are many ways to see that, at least for absolute values | x| < 1, Euler is right in that that arises while expanding the expression 1⁄ (1+ x) 2, which this series is indeed equal to after we set x = 1. The idea becomes clearer by considering the general series 1 − 2 x + 3 x 2 − 4 x 3 + 5 x 4 − 6 x 5 + &c.
is 1⁄ 4 since it arises from the expansion of the formula 1⁄ (1+1) 2, whose value is incontestably 1⁄ 4. it is no more doubtful that the sum of this series 1 − 2 + 3 − 4 + 5 etc. , his ideas are similar to what is now known as Abel summation: Įuler proposed a generalization of the word "sum" several times. But I have already noticed at a previous time, that it is necessary to give to the word sum a more extended meaning . For by adding 100 terms of this series, we get −50, however, the sum of 101 terms gives +51, which is quite different from 1⁄ 4 and becomes still greater when one increases the number of terms. when it is said that the sum of this series 1 − 2 + 3 − 4 + 5 − 6 etc. In a 1749 report, Leonhard Euler admits that the series diverges but prepares to sum it anyway: He formally defined the (C, n) methods in 1890 in order to state his theorem that the Cauchy product of a (C, n)-summable series and a (C, m)-summable series is (C, m + n + 1)-summable. , to 1⁄ 4 by a method that may be rephrased as (C, n) but was not justified as such at the time. In particular, he summed 1 − 2 + 3 − 4 +. 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 +.
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