Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31
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Grandi's series is Abel summable to 1/21/2 but its partial sums do not converge

Statement

Grandi's series 1−1+1−1+⋯1-1+1-1+\cdots is Abel summable to 1/21/2 but diverges ordinarily.

Facts & Assumptions

Given: Coefficients an=(−1)na_n=(-1)^n.

[L1]

For 0≤x<10\le x<1, ∑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]

Abel summability and ordinary convergence are defined through the boundary limit and partial sums, respectively (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).

Verification

technique · direct
1.1

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

L1L2
2.1

The partial sums alternate between 11 and 00, so they diverge. This concretely refutes FALSE: Abel summability alone implies ordinary convergence of a series.

givenL2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources