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For $a_n=(-1{)}^n(n+1)$, direct finite summation gives $s_{2m}=m+1$ and $s_{2m+1}=-(m+1)$. ConsequentlyThe distinct even and odd subsequential limits prevent first-order Cesàro summation. More averaging is a different method; a value assigned by another method is not a first-order Cesàro sum.

The finite limit of a sequence of averaged partial sums in the declared Cesàro summation convention, when that limit exists. It can exist when an ordinary sum does not, but every ordinarily convergent series has the same Cesàro sum as its ordinary sum.

Cesàro summation assigns a value to a series when the arithmetic means of its partial sums approach a finite limit of a sequence. Here we use the first-order method, often written $(C,1)$. For a real or complex series beginning at index zero, defineIt is Cesàro summable to $L$ when ${\sigma }_N\to L$. The averaging is applied to the partial sums $s_n$, rather than to the original terms $a_n$. This distinction is essential: the terms can average to zero while their accumulated sum keeps growing. ^[ref]

For Grandi's series $1-1+1-1+\cdots$, the partial sums alternate between one and zero. The series is not a convergent series, but the two values occupy asymptotically equal proportions of the averaging window. In the declared index convention,so its first-order Cesàro sum is $1/2$. This names a particular summation method; it does not turn the oscillating partial sums into an ordinarily convergent sequence.

Why is this useful? An oscillating approximation can retain stable information when its cumulative average settles. Because the Cesàro theorem for convergent sequences preserves an existing limit of a sequence, this adds some new cases without changing the sum of any ordinarily convergent series. For Fourier series, averaging the Fourier partial sums gives a Fejér sum; the existing Fejér kernel and approximation articles explain its effect on reconstruction. Averaging does not guarantee success for every oscillating sequence, as the bounded partial sums need not be Cesàro summable example shows.

One historical motivation was the multiplication of series. In 1918, A. F. Andersen recalled Ernesto Cesàro's product theorem and cited his 1890 publication at the start of a study of products involving Cesàro summable series. This connects the theory with preserving useful operations on sums, as well as with assigning a value to an isolated example. His paper treats several orders of summability; the examples here use only $(C,1)$.^[ref]

If a real or complex sequence converges to $a$, its Cesàro means have the same limit of a sequence. This is why Cesàro summation is a regular summation method when applied to the partial sums of a series.

Proof outline. Given $\epsilon >0$, choose $K$ so that $|a_k-a|<\epsilon$ for $k>K$. The triangle inequality gives, for $n>K$,The finite initial contribution tends to zero, and the remaining contribution is at most $\epsilon$. The converse fails: the alternating sequence $(-1{)}^n$ has Cesàro means tending to zero while its terms do not converge.

A series whose partial sums do not have a finite limit of a sequence. They may grow without bound or oscillate. Some divergent series belong to the domain of a summation method, such as Cesàro summation.

The series with terms $a_n=(-1{)}^n$, indexed from zero, named for Guido Grandi. Its even-indexed partial sums are one and its odd-indexed partial sums are zero, so it has no ordinary sum. First-order Cesàro summation gives $1/2$. Grouping adjacent pairs would obscure this distinction: the partial sums of the grouped expressions select only one of the original subsequences.

A summation method that includes every ordinarily convergent series and assigns it its ordinary sum. The Cesàro theorem for convergent sequences makes first-order Cesàro summation regular. Regularity alone does not justify every rearrangement or multiplication of a divergent series.

A series adds the terms of a sequence in their specified order. Its infinite-sum notation is defined through finite partial sums:when the displayed limit of a sequence exists as a finite number. Otherwise the series is not a convergent series in the ordinary sense. Writing an infinite list of additions does not by itself assign it a finite sum. Cesàro summation is a different prescription, with its own definition.

Two positive series make the distinction concrete. For $a_n=1/n$, the individual terms tend to zero but the harmonic series has unbounded partial sums. For $a_n=1/[n(n+1)]$, partial fraction decomposition reveals a telescoping series whose partial sums tend to one. The useful question is how much all the remaining terms can contribute together, rather than only how small the next term looks.

By A. F. Andersen. Published in Mathematisk-fysiske Meddelelser, volume I, number 4, Det Kgl. Danske Videnskabernes Selskab. The scanned title page names the author and gives Copenhagen and 1918.

Online text: gymarkiv.sdu.dk/MFM/kdvs/mfm%201-9/mfm-1-4.pdf

Cited passages: §1, printed p. 3, introduces orders of Cesàro summation and cites Ernesto Cesàro's 1890 publication in Bulletin des Sciences Mathématiques, series 2, volume 14, p. 114; §2, pp. 6–7, discusses the multiplication theorem and its role in developing summability theory. The original 1890 paper was not inspected here; the historical citation refers to Andersen's account.