False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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.

FALSE: convergence of a power series at one point other than its centre forces convergence at every real point

Statement

False claim: if a power series converges at one point distinct from its centre, then it converges at every real point.

Facts & Assumptions

Given: The geometric power series $\sum_{n\ge0}x^n$ centred at $0$ .

[L1]

The geometric series converges when $|x|<1$ and diverges when $|x|\ge1$ (For $|r| < 1$ , $\sum_{k \ge 0} r^k = 1/(1-r)$ , and for $|r| \ge 1$ the series diverges).

Refutation

technique · direct

1.1

At $x=1/2\ne0$ , the series converges by [L1]. At $x=2$ , its terms do not tend to zero and it diverges.

2.1

This single power series satisfies the premise and fails the conclusion, refuting the claim.

step 1.1∎

Depends on

For $|r| < 1$, $\sum_{k \ge 0} r^k = 1/(1-r)$, and for $|r| \ge 1$ the series diverges

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: 57 results over 16 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

Power series, Encyclopedia of Mathematics (standard reference, not scraped)