False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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FALSE: Abel summability alone implies ordinary convergence of a series

Statement

False claim: every Abel-summable real series converges ordinarily.

Facts & Assumptions

Given: Grandi's series $\sum_{n \geq 0} (- 1)^{n}$ .

[L1]

For $0 \leq x < 1$ , the geometric series gives $\sum_{n \geq 0} (- 1)^{n} x^{n} = 1 / (1 + x)$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[L2]

Ordinary convergence means convergence of the partial-sum sequence, whereas Abel summability uses the boundary limit of the power series (Series, partial sums, convergence and the sum, divergence, and the tail series, Abel summability by $\lim_{x ↑ 1} \sum a_{n} x^{n}$ and Cesaro summability by the Cesaro means of the partial sums).

Refutation

technique · direct

1.1

By [L1], the Abel transform tends to $1 / 2$ as $x ↑ 1$ , so the series is Abel summable to $1 / 2$ .

L1L2

2.1

Its inclusive partial sums alternate between $1$ and $0$ , so they do not converge. Hence Abel summability alone does not imply ordinary convergence.

givenL2∎

Depends on

Used by

Grandi's series is Abel summable to 1/2 but its partial sums do not converge Counterexample

Dependency tree · two levels

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Sources

Abel summability, Encyclopedia of Mathematics (standard reference, not scraped)
S. Semmes, Rice Math 322 notes (standard reference, not scraped)
Tauberian theorems, Encyclopedia of Mathematics (standard reference, not scraped)