Cesàro summation

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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. However, a series which postively diverges to an infinite value, will not be summable to a finite value in any case.

Cesàro summation is named for the Italian analyst Ernesto Cesàro (1859–1906).

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[edit] 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 sequence {an} is called Cesàro summable, with Cesàro sum A, if

\lim_{n\to\infty} \frac{s_1 + \cdots + s_n}{n} = A.

[edit] 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.

[edit] (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; 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

\lim_{n\to\infty}\frac{A_n^\alpha}{E_n^\alpha}

if it exists (Shawyer & Watson 1994, pp.16-17).

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 Σan 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.

[edit] 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 0}\int_0^{1/\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 0}\frac{1}{\lambda}\int_0^\lambda\left\{\int_0^xf(y)\, dy\right\}dx

which is the limit of means of the partial integrals.

[edit] See also

[edit] References