The Cesaro means $\sigma_n = (x_0 + \dots + x_n)/(n+1)$ and $(C,1)$-summability

Let $(x_k)$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences), with finite sums as in Finite sums and finite products, by recursion. For $n \in \mathbb{N}$ the $n$-th Cesaro mean of $(x_k)$ is

$\sigma_n \;:=\; \frac{1}{n+1}\sum_{k=0}^{n} x_k \;=\; \frac{x_0 + x_1 + \dots + x_n}{n+1},$

where $n+1$ denotes the canonical natural $(n+1)\cdot 1_{\mathbb{R}}$.

This is well defined. The only thing that could fail is the division: since $n + 1 \ge 1$, the canonical natural $(n+1)\cdot 1_{\mathbb{R}}$ is strictly positive (Canonical naturals are positive and strictly increasing), hence nonzero by trichotomy (Complete ordered field (least-upper-bound property)), hence invertible. The sum $\sum_{k=0}^{n} x_k$ is the finite sum $\sum_{k < n+1} x_k$ of Finite sums and finite products, by recursion, a single well-determined real for each $n$. So $(\sigma_n)_{n \in \mathbb{N}}$ is again a sequence of reals.

The sequence $(x_k)$ is $(C,1)$-summable to $L \in \mathbb{R}$, or Cesaro summable to $L$, when the sequence of Cesaro means converges to $L$ (Limits and Cauchy sequences of reals). Limits of real sequences are unique (A sequence has at most one limit), so such an $L$ is unique when it exists, and we write it $\lim_n \sigma_n$.

• The indexing starts at $0$ and the denominator is $n+1$, not $n$. Sequences here are functions on $\mathbb{N}$ and $\mathbb{N}$ contains $0$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences), so $\sigma_n$ averages the $n+1$ terms $x_0, \dots, x_n$. Writing $\sigma_n = (x_1 + \dots + x_n)/n$, as texts indexing from $1$ do, would leave $\sigma_0$ undefined and would not be a sequence on $\mathbb{N}$ at all. The convention chosen here is also what makes the Cesaro matrix $c_{n,k} = 1/(n+1)$ for $k \le n$ a genuine $\mathbb{N} \times \mathbb{N}$ summability matrix (A summability (Toeplitz) matrix, the transformed sequence $y_n = \sum_k c_{n,k} x_k$, and regularity, The Cesaro matrix satisfies the Silverman-Toeplitz conditions, giving a second proof of the Cesaro mean theorem).

• $(C,1)$-summability is strictly weaker than convergence. Every convergent sequence is $(C,1)$-summable to its limit (If $x_k \to L$ then $\sigma_n \to L$: convergence implies $(C,1)$-summability to the same value), and the converse fails (FALSE: if the Cesaro means of a sequence converge then the sequence converges): the Cesaro means of the alternating sequence converge to $0$ while the sequence itself diverges (The Cesaro means of $(-1)^k$ converge to $0$ although the sequence diverges ↗). That is the entire point of the notion: it assigns a value to some divergent sequences, consistently with the ordinary limit wherever the ordinary limit exists.

• Nothing here sums a series. The object averaged is the sequence $(x_k)$ itself, not its partial sums, and $\sum_{k=0}^{n} x_k$ is a finite sum in the sense of Finite sums and finite products, by recursion. Applied instead to the partial sums of a series, the same definition gives the classical Cesaro summation of divergent series; series are not available until the next page of this track and nothing above presupposes them.