= Cesàro summation
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{title2=$\sigma_N=(N+1)^{-1}\sum_{n=0}^{N}s_n$}
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= Cesàro summable
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<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, define
$$
s_n=\sum_{k=0}^n a_k,\qquad
\sigma_N=\frac1{N+1}\sum_{n=0}^{N}s_n.
$$
It 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. <Cesàro Summation (Mathematica tutorial) by Vladimir Dobrushkin>{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,
$$
\sigma_{2m}=\frac{m+1}{2m+1},\qquad
\sigma_{2m+1}=\frac12,
$$
so \b[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>.
\Image[/cesaro-summation-alternating-series.png]
{title=<Cesàro summation> succeeds for <Grandi's series>, but first-order averaging fails for $1-2+3-4+\cdots$}
{description=Original exact <partial sums> and <Cesàro means>, with the first term indexed by zero. In the second row, the even and odd <subsequences> of the averages have different limits.}
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 $(C,1)$.<Sur la multiplication de séries absolument convergentes par des séries sommables par la méthode de Cesàro by A. F. Andersen>[§1, footnote, p. 3; §2, pp. 6–7]{ref}
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