TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 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.

Frobenius' theorem: Cesaro summability of a real series implies Abel summability to the same value

Statement

If a real series is Cesaro summable to $s$ , then it is Abel summable to $s$ .

Facts & Assumptions

Given: Cesaro means $σ_{n} \to s$ for the partial sums of $\sum a_{n}$ .

[L1]

If $(σ_{n})$ is bounded, the Abel series converges for $0 < x < 1$ and its transform is $A (x) = (1 - x)^{2} \sum_{n \geq 0} ι (n + 1) σ_{n} x^{n}$ (For $0 < x < 1$ , the Abel transform of a series is $(1 - x)^{2} \sum_{n \geq 0} (n + 1) σ_{n} x^{n}$ , where $σ_{n}$ are the Cesaro means of its partial sums).

[L2]

The nonnegative weights $w_{n} (x) : = (1 - x)^{2} ι (n + 1) x^{n}$ sum to $1$ for $0 < x < 1$ . Indeed, apply the transform in [L1] to the series with coefficients $1, 0, 0, \dots$ , whose partial sums and Cesaro means are all $1$ (For $0 < x < 1$ , the Abel transform of a series is $(1 - x)^{2} \sum_{n \geq 0} (n + 1) σ_{n} x^{n}$ , where $σ_{n}$ are the Cesaro means of its partial sums).

[L3]

Abel summability to $s$ means that the Abel series converges on $0 \leq x < 1$ and tends to $s$ as $x ↑ 1$ (Abel summability by $\lim_{x ↑ 1} \sum a_{n} x^{n}$ and Cesaro summability by the Cesaro means of the partial sums).

Proof

technique · direct

1.1

Since $(σ_{n})$ converges, it is bounded, so [L1] applies and $A (x) - s = \sum_{n \geq 0} w_{n} (x) (σ_{n} - s)$ .

givenL1L2

2.1

Given $ε > 0$ , choose $N$ with $∣ σ_{n} - s ∣ < ε$ for $n \geq N$ . The corresponding tail is at most $ε \sum_{n \geq N} w_{n} (x) \leq ε$ .

step 1.1L2choose

3.1

For each fixed $n$ , $w_{n} (x) \to 0$ as $x ↑ 1$ , so the finite head tends to $0$ . Together with step 2.1 this gives $A (x) \to s$ , which is Abel summability by definition.

step 1.1step 2.1L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cesàro summation, Encyclopedia of Mathematics (standard reference, not scraped)
Cesàro summation methods, Encyclopedia of Mathematics (standard reference, not scraped)
S. Semmes, Rice Math 322 notes (standard reference, not scraped)