Alphabeta Math
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: every power series converges uniformly on its entire open interval of convergence

Statement

False claim: every real power series converges uniformly on its entire open interval of convergence.

Facts & Assumptions

Given: The geometric power series ∑n≥0xn\sum_{n\ge0}x^n on (−1,1)(-1,1).

[L1]

It converges pointwise there to 1/(1−x)1/(1-x) (For ∣r∣<1|r| < 1, ∑k≥0rk=1/(1−r)\sum_{k \ge 0} r^k = 1/(1-r), and for ∣r∣≥1|r| \ge 1 the series diverges).

[L2]

A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy (A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy).

Refutation

technique · direct
1.1

For every NN, the difference between the (N+1)(N+1)st and NNth partial sums is xNx^N. Its supremum over x∈(−1,1)x\in(-1,1) is 11.

givenalgebra
2.1

Thus the partial sums are not uniformly Cauchy and cannot converge uniformly by [L2], despite pointwise convergence on the entire open radius interval by [L1].

step 1.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 67 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