ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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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Abel convergence of the alternating harmonic power series along a nonradial Stolz approach

Example

For $k \geq 2$ , put $z_{k} = 1 - 1 / k + i / (2 k)$ . Then $∣ z_{k} ∣ < 1$ , $z_{k} \to 1$ nonradially within one Stolz region, and $\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1} z_{k}^{n}}{n} ⟶ \log 2.$

Facts & Assumptions

Given: The sequence $(z_{k})$ and the alternating harmonic coefficients.

[L1]

The alternating harmonic series converges to $\log 2$ ([The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series)).

[L2]

Abel's theorem recovers a convergent series along every fixed Stolz approach to $1$ (Abel's limit theorem: a convergent complex series is recovered by its power series along every Stolz approach to 1).

Verification

technique · direct

1.1algebra

Direct calculation gives $∣ z_{k} ∣^{2} = 1 - 2 / k + 5 / (4 k^{2}) < 1$ for $k \geq 2$ , and $∣ 1 - z_{k} ∣ = \sqrt{5} / (2 k)$ .

2.1step 1.1algebra

Since $1 - ∣ z_{k} ∣ = (1 - ∣ z_{k} ∣^{2}) / (1 + ∣ z_{k} ∣)$ , step 1.1 gives a uniform bound on $∣ 1 - z_{k} ∣ / (1 - ∣ z_{k} ∣)$ , so $(z_{k})$ lies in one Stolz region and tends to $1$ .

3.1step 2.1L1L2∎

Apply [L2] to the series in [L1]. The nonzero imaginary part makes the approach nonradial.

Depends on

Used by

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Dependency tree · two levels

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