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Cesàro summation : ウィキペディア英語版
Cesàro summation

In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not convergent in the usual sense, while coinciding with the standard sum if they are convergent. 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).
== Definition ==
Let be a sequence, and let
:s_k = a_1 + \cdots + a_k
be the ''k''th partial sum of the series
:\sum_^\infty a_n.
The series \sum_^\infty 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:
:\lim_ \frac\sum_^n s_k = 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. It is easy to show that any convergent series is Cesàro summable, and the sum of the series agrees with its Cesàro sum. However, as the first example below demonstrates, there are series that diverge but are nonetheless Cesàro summable.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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