Cesàro summation

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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

$s_k = a_1 + \cdots + a_k$

be the kth partial sum of the series

$\sum_{n=1}^\infty a_n.$

The series {sn} is called Cesàro summable, with Cesàro sum $A \in \R$, if the average value of its partial sums tends to A:

$\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n s_k = A.$

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 an = (-1)n+1 for n ≥ 1. That is, {an} is the sequence

$1, -1, 1, -1, \ldots.\,$

Then the sequence of partial sums {sn} is

$1, 0, 1, 0, \ldots,\,$

so that the series, known as Grandi's series, clearly does not converge. On the other hand, the terms of the sequence {(s1 + ... + sn)/n} are

$\frac{1}{1}, \,\frac{1}{2}, \,\frac{2}{3}, \,\frac{2}{4}, \,\frac{3}{5}, \,\frac{3}{6}, \,\frac{4}{7}, \,\frac{4}{8}, \,\ldots,$

so that

$\lim_{n\to\infty} \frac{s_1 + \cdots + s_n}{n} = 1/2.$

Therefore the Cesàro sum of the sequence {an} is 1/2.

On the other hand, let an = 1 for n ≥ 1. That is, {an} is the sequence

$1, 1, 1, 1, \ldots.\,$

Then the sequence of partial sums {sn} is

$1, 2, 3, 4, \ldots,\,$

and the series diverges to infinity. The terms of the sequence {(s1 + ... + sn)/n} are

$\frac{1}{1}, \,\frac{3}{2}, \,\frac{6}{3}, \,\frac{10}{4}, \,\ldots.$

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 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

$A_n^{-1}=a_n;\quad A_n^\alpha=\sum_{k=0}^n A_k^{\alpha-1}$

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

$(C,\alpha)-\sum_{j=0}^\infty a_j=\lim_{n\to\infty}\frac{A_n^\alpha}{E_n^\alpha}$

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

$(C,\alpha)-\sum_{j=0}^\infty a_j = \lim_{n\to\infty} \sum_{j=0}^n \frac{{n \choose j}}{{n+\alpha \choose j}} a_j.$

Even more generally, for $\alpha\in\mathbb{R}\setminus(-\mathbb{N})$, let Anα be implicitly given by the coefficients of the series

$\sum_{n=0}^\infty A_n^\alpha x^n=\frac{\displaystyle{\sum_{n=0}^\infty a_nx^n}}{(1-x)^{1+\alpha}},$

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 $\scriptstyle{\int_0^\infty f(x)\,dx}$ is Cesàro summable (C, α) if

$\lim_{\lambda\to\infty}\int_0^\lambda\left(1-\frac{x}{\lambda}\right)^\alpha f(x)\, dx$

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

$\lim_{\lambda\to \infty}\frac{1}{\lambda}\int_0^\lambda\left\{\int_0^xf(y)\, dy\right\}\,dx$

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.

References

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