Cesàro summation

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In mathematical analysis, Cesàro summation (also known as the Cesàro mean[1][2]) assigns values to some infinite sums that are not convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.

Cesàro summation is named for the Italian analyst Ernesto Cesàro (1859–1906). It is a special case of a matrix summability method.

The term summation can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the Eilenberg–Mazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that the sum of that series is 1/2.

Definition[edit]

Let be a sequence, and let

be its kth partial sum.

The sequence (an) is called Cesàro summable, with Cesàro sum A ∈ ℝ, if, as n tends to infinity, the arithmetic mean of its first n partial sums s1, s2, ..., sn tends to A:

The value of the resulting limit is called the Cesàro sum of the series If this series is (conditionally) convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.

Examples[edit]

First example[edit]

Let an = (−1)n for n ≥ 0. That is, is the sequence

Let G denote the series

The series G is known as Grandi's series.

Let denote the sequence of partial sums of G:

This sequence of partial sums does not converge, so the series G is divergent. However, G is Cèsaro summable. Let be the sequence of arithmetic means of the first n partial sums:

Then

and therefore, the Cesàro sum of the series G is 1/2.

Second example[edit]

As another example, let an = n for n ≥ 1. That is, is the sequence

Let G now denote the series

Then the sequence of partial sums is

Since the sequence of partial sums grows without bound, the series G diverges to infinity. The sequence (tn) of means of partial sums of G is

This sequence diverges to infinity as well, so G is not Cesàro summable. In fact, for any sequence 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[edit]

In 1890, Ernesto Cesàro stated a broader family of summation methods which have since been called (C, α) for non-negative integers α. 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 an, define the quantities

(where the upper indices do not denote exponents) and define Eα
n
to be Aα
n
for the series 1 + 0 + 0 + 0 + …. Then the (C, α) sum of an is denoted by (C, α)-∑an and has the value

if it exists (Shawyer & Watson 1994, pp.16-17). This description represents an α-times iterated application of the initial summation method and can be restated as

Even more generally, for α ∈ ℝ \ ℤ, let Aα
n
be implicitly given by the coefficients of the series

and Eα
n
as above. In particular, Eα
n
are the binomial coefficients of power −1 − α. Then the (C, α) sum of an is defined as above.

If an has a (C, α) sum, then it also has a (C, β) sum for every β > α, and the sums agree; furthermore we have an = o(nα) if α > −1 (see little-o notation).

Cesàro summability of an integral[edit]

Let α ≥ 0. The integral is (C, α) summable 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[edit]

References[edit]

  1. ^ Hardy, G. H. (1992). Divergent Series. Providence: American Mathematical Society. ISBN 978-0-8218-2649-2.
  2. ^ Katznelson, Yitzhak (1976). An Introduction to Harmonic Analysis. New York: Dover Publications. ISBN 978-0-486-63331-2.