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 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).

Definition

Let {a_n} be a sequence, and let

<math>s_k = a_1 + \cdots + a_k</math>

be the kth partial sum of the series

<math>\sum_{n=1}^\infty a_n</math>.

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

<math>\lim_{n\to\infty} \frac{s_1 + \cdots + s_n}{n} = \alpha</math>.

Examples

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

<math>1, -1, 1, -1, \ldots</math>.

Then the sequence of partial sums {s_n} is

<math>1, 0, 1, 0, \ldots</math>,

so that the series clearly does not converge. On the other hand, the terms of the sequence {(s₁ + ... + s_n)/n} are

<math>\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</math>,

so that

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

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

See also

• Cesàro mean
• Abel summation
• Borel summation
• Divergent series