Cesàro summation

In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not convergent in the usual sense. The Cesàro sum is defined as the limit of the arithmetic mean of the partial sums of the series.

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

The term summation can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the Eilenberg–Mazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that the sum of that series is $1 / 2$ , a result that can readily be disproven.

Let ${a n}$ be a sequence, and let

be the $k$ th partial sum of the series

The series $\sum \infty n = 1 a n$ is called Cesàro summable, with Cesàro sum $A \in ℝ$ , if the average value of its partial sums $s k$ tends to $A$ :

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. If a series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum. For any convergent sequence, the corresponding series is Cesàro summable and the limit of the sequence coincides with the Cesàro sum.

Let $a n = (-1) n$ for $n \geq 0$ . That is, ${a n}$ is the sequence

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