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CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The log(1+x) power series diverges at x=-1

Statement refuted

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

Facts & Assumptions

Given: The endpoint x=−1x=-1.

[L1]

The formula for log⁡(1+x)\log(1+x) is stated only on (−1,1](-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\sum_{n\ge1}1/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=−1x=-1, its nn-th formal term is (−1)n+1(−1)n/n=−1/n(-1)^{n+1}(-1)^n/n=-1/n.

givenalgebra
2.1

Hence the formal series is −∑n≥11/n-\sum_{n\ge1}1/n, which diverges by [L2].

step 1.1L2
3.1

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

step 2.1L1∎

Depends on

Used by

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Sources