Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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FALSE: convergence of a complex power series at one point other than its centre forces convergence everywhere

Statement

False claim. If a complex power series converges at one point other than its centre, then it converges at every complex point.

Facts & Assumptions

Given: The geometric series ∑n≥0zn centred at 0.

[L1]

Cauchy–Hadamard gives absolute convergence inside the radius, divergence outside it, and no boundary assertion (Cauchy-Hadamard for complex power series, including zero and infinite radius).

[L2]

If the terms of a real series do not tend to 0, that real series diverges (If a series converges then its terms tend to 0).

Refutation

technique · direct
1.1

At z=1/2, the finite geometric identity shows that the partial sums tend to 2, so the series converges at a noncentral point.

algebra
1.2

At z=2, its terms 2n do not tend to 0, so the series diverges by [L2].

L2
2.1

Hence the claim is false. Consistently, [L1] gives this series radius 1: one interior convergence point does not force an infinite radius.

step 1.1step 1.2L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 76 results over 15 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