CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Abel summability does not imply ordinary convergence

Statement refuted

If a series is Abel summable, then its ordinary partial sums converge.

Facts & Assumptions

Given: Grandi's series $1 - 1 + 1 - 1 + \dots .$

[F1]

For $∣ r ∣ < 1$ , the geometric-series identity gives $1 - r + r^{2} - r^{3} + \dots = \frac{1}{1 + r} .$

Counterexample

technique · direct

1.1givenalgebra

The ordinary partial sums are $1, 0, 1, 0, \dots,$ so they do not converge.

2.1F1step 1.1algebra∎

For $0 \leq r < 1$ , [F1] makes the associated Abel sum $1 - r + r^{2} - r^{3} + \dots = \frac{1}{1 + r} .$ As $r ↑ 1$ , this tends to $1 / 2$ . Thus the series is Abel summable but not ordinarily convergent, refuting the statement.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Grandi's series (Wikipedia) (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)