Sequences, series, summability Definition

Cesàro mean

Definition

Let $(a_n)_{n \ge 1}$ be a real sequence. For each integer $n \ge 1$ , the $n$ -th Cesàro mean of $(a_n)$ is the arithmetic mean of its first $n$ terms, $\sigma_n = \frac{a_1 + a_2 + \cdots + a_n}{n}.$

Remarks

The sequence $(-1)^{n+1}$ diverges, but its Cesàro means are $1, 0, 1/3, 0, 1/5, \dots$ , and they tend to $0$ . If $(a_n)$ converges, the Cesàro means converge to the same limit. Cesàro summation applies the Cesàro means to the partial sums of a series. A series is Cesàro summable with sum $A$ when the means of its partial sums tend to $A$ .

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Sequence definition
Real sequence definition
Cesàro mean definition

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Tag	`0P0DJHBGBB`
URI	`m2://0P0DJHBGBB`
Kind	definition
Writing	W1 stated
MSC	40G05
Defines	Cesàro mean
Uses	Real sequence

Cesàro mean

Definition

Remarks

Reading path

Where this leads

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