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Concept# Cesàro summation

Summary

In mathematical analysis, Cesàro summation (also known as the Cesàro mean) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
This special case of a matrix summability method 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.
Let be a sequence, and let
be its kth partial sum.
The sequence (an) is called Cesàro summable, with Cesàro sum A ∈ , if, as n tends to infinity, the arithmetic mean of its first n partial sums s1, s2, ..., sn tends to A:
The value of the resulting limit is called the Cesàro sum of the series If this series is convergent, then it is Cesàro summable and its Cesàro sum is the usual sum.
Let an = (−1)n for n ≥ 0. That is, is the sequence
Let G denote the series
The series G is known as Grandi's series.
Let denote the sequence of partial sums of G:
This sequence of partial sums does not converge, so the series G is divergent. However, G Cesàro summable. Let be the sequence of arithmetic means of the first n partial sums:
Then
and therefore, the Cesàro sum of the series G is 1/2.
As another example, let an = n for n ≥ 1. That is, is the sequence
Let G now denote the series
Then the sequence of partial sums is
Since the sequence of partial sums grows without bound, the series G diverges to infinity. The sequence (tn) of means of partial sums of G is
This sequence diverges to infinity as well, so G is Cesàro summable. In fact, for any sequence which diverges to (positive or negative) infinity, the Cesàro method also leads to a sequence that diverges likewise, and hence such a series is not Cesàro summable.

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Divergent series

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.

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