ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

A power series with reciprocal-square coefficients converges uniformly on the closed unit disc

Example

The series $\sum_{n = 1}^{\infty} z^{n} / n^{2}$ converges absolutely and uniformly on the closed unit disc $∣ z ∣ \leq 1$ .

Facts & Assumptions

Given: A complex number $z$ with $∣ z ∣ \leq 1$ .

[L1]

The complex M-test gives absolute pointwise and uniform convergence under a convergent nonnegative majorant (Weierstrass M-test for complex-valued function series).

[L2]

For real $p$ , the real series $\sum_{n \geq 1} 1 / n^{p}$ converges exactly when $p > 1$ (The p-series for a real exponent p converges exactly when p is greater than one).

Verification

technique · direct

1.1givenalgebra

For $n \geq 1$ , $∣ z^{n} / n^{2} ∣ \leq 1 / n^{2}$ .

2.1step 1.1L1L2∎

The majorant converges by [L2], so [L1] proves absolute and uniform convergence on the entire closed disc, including its boundary.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.