Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The alternating harmonic series sums to log 2

Example

∑n=1∞(−1)n+1n=log⁡2.\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}n=\log2.

Facts & Assumptions

Given: The endpoint formula for log⁡(1+x)\log(1+x).

[L1]

At x=1x=1, log⁡(1+x)=∑n≥1(−1)n+1xn/n\log(1+x)=\sum_{n\ge1}(-1)^{n+1}x^n/n converges to log⁡2\log2 ([The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series)).

Verification

technique · direct
1.1

Substituting x=1x=1 into [L1] gives exactly the displayed alternating harmonic series.

L1
2.1

Therefore the alternating harmonic series has sum log2log2.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 46 results over 10 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