ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

The alternating harmonic power series tends to log 2 at the boundary point 1

Example

For $∣ z ∣ < 1$ , put $F (z) = \sum_{n = 1}^{\infty} (- 1)^{n + 1} z^{n} / n$ . Then $F (z) \to \log 2$ as $z \to 1$ within any fixed Stolz region.

Facts & Assumptions

Given: The alternating harmonic coefficients indexed from $n = 1$ .

[L1]

The alternating harmonic series converges to $\log 2$ ([The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series)).

[L2]

A convergent complex series is recovered by its power series along every Stolz approach to $1$ (Abel's limit theorem: a convergent complex series is recovered by its power series along every Stolz approach to 1).

Verification

technique · direct

1.1

Regard the real coefficients $(- 1)^{n + 1} / n$ as complex coefficients; by [L1] their series has sum $\log 2$ .

2.1

Apply [L2] to obtain the asserted angular limit. The claim concerns only the boundary limit and introduces no logarithm branch inside the disc.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2, Abel's theorem (standard reference, not scraped)