 Cesàro summation

In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A. The significance of Cesàro summation is that a series which does not converge may still have a welldefined Cesàro sum.
Cesàro summation is named for the Italian analyst Ernesto Cesàro (1859–1906).
Contents
Definition
Let {a_{n}} be a sequence, and let
be the kth partial sum of the series
The series {s_{n}} is called Cesàro summable, with Cesàro sum , if the average value of its partial sums tends to A:
In other words, the Cesàro sum of an infinite series is the limit of the arithmetic mean (average) of the first n partial sums of the series, as n goes to infinity.
Examples
Let a_{n} = (1)^{n+1} for n ≥ 1. That is, {a_{n}} is the sequence
Then the sequence of partial sums {s_{n}} is
so that the series, known as Grandi's series, clearly does not converge. On the other hand, the terms of the sequence {(s_{1} + ... + s_{n})/n} are
so that
Therefore the Cesàro sum of the sequence {a_{n}} is 1/2.
On the other hand, let a_{n} = 1 for n ≥ 1. That is, {a_{n}} is the sequence
Then the sequence of partial sums {s_{n}} is
and the series diverges to infinity. The terms of the sequence {(s_{1} + ... + s_{n})/n} are
Thus, this sequence also diverges to infinity, and the series is not Cesàro summable. More generally, for a series which diverges to (positive or negative) infinity the Cesàro method leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable. Since a sequence that is ultimately monotonic either converges or diverges to infinity, it follows that a series which is not convergent but Cesàro summable oscillates.
(C, α) summation
In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called (C, n) for nonnegative integers n. The (C, 0) method is just ordinary summation, and (C, 1) is Cesàro summation as described above.
The higherorder methods can be described as follows: given a series Σa_{n}, define the quantities
and define E_{n}^{α} to be A_{n}^{α} for the series 1 + 0 + 0 + 0 + · · ·. Then the (C, α) sum of Σa_{n} is denoted by (C, α)Σa_{n} and has the value
if it exists (Shawyer & Watson 1994, pp.1617). This description represents an αtimes iterated application of the initial summation method and can be restated as
Even more generally, for , let A_{n}^{α} be implicitly given by the coefficients of the series
and E_{n}^{α} as above. In particular, E_{n}^{α} are the binomial coefficients of power −1 − α. Then the (C, α) sum of Σ a_{n} is defined as above.
The existence of a (C, α) summation implies every higher order summation, and also that a_{n} = o(n^{α}) if α > −1.
Cesàro summability of an integral
Let α ≥ 0. The integral is Cesàro summable (C, α) if
exists and is finite (Titchmarsh 1948, §1.15). The value of this limit, should it exist, is the (C, α) sum of the integral. Analogously to the case of the sum of a series, if α=0, the result is convergence of the improper integral. In the case α=1, (C, 1) convergence is equivalent to the existence of the limit
which is the limit of means of the partial integrals.
As is the case with series, if an integral is (C,α) summable for some value of α ≥ 0, then it is also (C,β) summable for all β > α, and the value of the resulting limit is the same.
See also
 Abel summation
 Borel summation
 Euler summation
 Cesàro mean
 Divergent series
 Fejér's theorem
 Riesz mean
 Abelian and tauberian theorems
 Silverman–Toeplitz theorem
 Summation by parts
References
 Shawyer, Bruce; Watson, Bruce (1994), Borel's Methods of Summability: Theory and Applications, Oxford UP, ISBN 0198535856.
 Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.), New York, N.Y.: Chelsea Pub. Co. (published 1986), ISBN 9780828403245.
 Volkov, I.I. (2001), "Cesàro summation methods", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/c/c021360.htm
 Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 9780521358859.
Categories: Summability methods
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