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CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02
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  • 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.
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The log(1+x) power series diverges at x=-1

Statement refuted

The power series for log⁡(1+x) converges at x=−1.

Facts & Assumptions

Given: The endpoint x=−1.

[L1]

The formula for log⁡(1+x) is stated only on (−1,1] ([The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series)).

[L2]

The series ∑n≥11/n diverges (The p-series for a real exponent p converges exactly when p is greater than one).

Counterexample

technique · direct
1.1

At x=−1, its n-th formal term is (−1)n+1(−1)n/n=−1/n.

givenalgebra
2.1

Hence the formal series is −∑n≥11/n, which diverges by [L2].

step 1.1L2
3.1

Thus x=−1 cannot be added to the convergence interval in [L1].

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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