Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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)=∑n=1∞(−1)n+1zn/n. Then F(z)→log⁡2 as z→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
1.1L1

Regard the real coefficients (−1)n+1/n as complex coefficients; by [L1] their series has sum log⁡2.

2.1step 1.1L2∎

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

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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