Cesàro summation

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For the song "Cesaro Summability", see Ænima.

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 α, then the series is also Cesàro summable and has Cesàro sum α. The significance of Cesàro summation is that a series which diverges may still have a well-defined Cesàro sum.

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

1 Definition
2 Examples
3 Generalizations
4 See also
5 Notes
6 References

[edit] Definition

Let {a_n} 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 {a_n} is called Cesàro summable, with Cesàro sum α, if

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

[edit] Examples

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

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

Then the sequence of partial sums {s_n} 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 + ... + s_n)/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 {a_n} is 1/2.

[edit] Generalizations

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 Σa_n, define the quantities

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