DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Abel summability by $\lim_{x ↑ 1} \sum a_{n} x^{n}$ and Cesaro summability by the Cesaro means of the partial sums

For a real series $\sum_{n \geq 0} a_{n}$ , write its inclusive partial sums as

$S_{n} : = \sum_{k = 0}^{n} a_{k}$

and their Cesaro means as

$σ_{n} : = \frac{1}{ι (n + 1)} \sum_{k = 0}^{n} S_{k} .$

It is Abel summable to $s$ if the power series

$A (x) : = \sum_{n = 0}^{\infty} a_{n} x^{n}$

converges for every $0 \leq x < 1$ and $\lim_{x ↑ 1} A (x) = s$ in the one-sided sense of The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ . These are summability methods for the zero-indexed series of Series, partial sums, convergence and the sum, divergence, and the tail series; they do not assert ordinary convergence.

Depends on