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 well-defined Cesàro sum.
Cesàro summation is named for the Italian analyst Ernesto Cesàro (1859–1906).
Definition
Let {an} be a sequence, and let
be the kth partial sum of the series
The series is called Cesàro summable, with Cesàro sum , if the average value of its partial sums tends to :
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. It is easy to show that any convergent series is Cesaro summable, and the sum of the series agrees with its Cesaro sum. However, as the first example below demonstrates, there are series that diverge but are nonetheless Cesaro summable.
Examples
Let an = (−1)n+1 for n ≥ 1. That is, {an} is the sequence
and let G denote the series
Then the sequence of partial sums {sn} is
so that the series G, known as Grandi's series, clearly does not converge. On the other hand, the terms of the sequence {tn} of the (partial) means of the {sn} where
are
so that
Therefore the Cesàro sum of the series G is 1/2.
On the other hand, now let an = n for n ≥ 1. That is, {an} is the sequence
and let G now denote the series
Then the sequence of partial sums {sn} is
and the evaluation of G diverges to infinity. The terms of the sequence of means of partial sums {tn } are here
Thus, this sequence diverges to infinity as well as G, and G is now not Cesàro summable. In fact, any series which diverges to (positive or negative) infinity the Cesàro method also leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable.
(C, α) summation
In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called (C, n) for non-negative integers n. The (C, 0) method is just ordinary summation, and (C, 1) is Cesàro summation as described above.
The higher-order methods can be described as follows: given a series Σan, define the quantities
and define Enα to be Anα for the series 1 + 0 + 0 + 0 + · · ·. Then the (C, α) sum of Σan is denoted by (C, α)-Σan and has the value
if it exists (Shawyer & Watson 1994, pp.16-17). This description represents an -times iterated application of the initial summation method and can be restated as
Even more generally, for , let Anα be implicitly given by the coefficients of the series
and Enα as above. In particular, Enα are the binomial coefficients of power −1 − α. Then the (C, α) sum of Σ an is defined as above.
The existence of a (C, α) summation implies every higher order summation, and also that an = o(nα) 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
- Abel's summation formula
- Abel–Plana formula
- Borel summation
- Euler summation
- Lambert summation
- Cesàro mean
- Divergent series
- Fejér's theorem
- Riesz mean
- Perron's formula
- 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 0-19-853585-6.
- Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.), New York, N.Y.: Chelsea Pub. Co. (published 1986), ISBN 978-0-8284-0324-5.
- Volkov, I.I. (2001), "Cesàro summation methods", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0-521-35885-9.