Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 ∑n≥0(−1)n\sum_{n\ge0}(-1)^n.

[L1]

For 0≤x<10\le x<1, the geometric series gives ∑n≥0(−1)nxn=1/(1+x)\sum_{n\ge0}(-1)^nx^n=1/(1+x) (For ∣r∣<1|r| < 1, ∑k≥0rk=1/(1−r)\sum_{k \ge 0} r^k = 1/(1-r), and for ∣r∣≥1|r| \ge 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∑anxn\lim_{x\uparrow1}\sum a_nx^n and Cesaro summability by the Cesaro means of the partial sums).

Refutation

technique · direct
1.1

By [L1], the Abel transform tends to 1/21/2 as x↑1x\uparrow1, so the series is Abel summable to 1/21/2.

L1L2
2.1

Its inclusive partial sums alternate between 11 and 00, so they do not converge. Hence Abel summability alone does not imply ordinary convergence.

givenL2∎

Depends on

Used by

Dependency tree · next 3 levels

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Sources