Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

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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 ∑n≥0xn centred at 0.

[L1]

The geometric series converges when ∣x∣<1 and diverges when ∣x∣≥1 (For ∣r∣<1, ∑k≥0rk=1/(1−r), and for ∣r∣≥1 the series diverges).

Refutation

technique · direct
1.1

At x=1/2≠0, the series converges by [L1]. At x=2, its terms do not tend to zero and it diverges.

L1
2.1

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

step 1.1∎

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