Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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The complex geometric series is not uniformly convergent on its open unit disc

Statement refuted

Every complex power series converges uniformly on its entire open disc of convergence.

Facts & Assumptions

Given: The geometric power series ∑n≥0zn on D={∣z∣<1}.

[L1]

Uniform convergence requires one index to work for every point of the domain (Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary).

[L2]

Cauchy–Hadamard gives the geometric series radius 1 and pointwise convergence for ∣z∣<1 (Cauchy-Hadamard for complex power series, including zero and infinite radius).

Counterexample

technique · direct
1.1

For each n≥1, sup⁡z∈D∣zn∣=1, although the supremum is not attained: real z↑1 makes zn↑1.

algebra
2.1

If the series converged uniformly, its terms would tend uniformly to 0, contradicting step 1.1 and the quantifiers in [L1].

step 1.1L1
3.1

Nevertheless [L2] gives pointwise convergence throughout D, so this is a counterexample to uniform convergence on the whole open disc.

step 2.1L2∎

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: 61 results over 14 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