Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The alternating harmonic series illustrates Abel's boundary-limit theorem without evaluating its sum
Statement
Let denote the ordinary sum of the alternating harmonic series
Then, without evaluating ,
Facts & Assumptions
Given: The zero-indexed alternating harmonic series.
It converges by the alternating-series test (The alternating series test: if is nonincreasing with then converges, the sum lies between any two consecutive partial sums, and the error after terms is at most , The canonical natural of a field).
Abel's limit theorem identifies the boundary limit with the ordinary sum of any convergent series (Abel's limit theorem: if a real series converges to , then its power series tends to as ).
Verification
By [L1], the ordinary sum exists.
Apply [L2] to its coefficients to obtain the displayed limit. No closed-form evaluation of is needed.
Depends on
- The alternating series test: if $(b_k)$ is nonincreasing with $b_k \to 0$ then $\sum_{k} (-1)^{k} b_k$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_n$
- Abel's limit theorem: if a real series converges to $s$, then its power series tends to $s$ as $x\uparrow1$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
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: 68 results over 14 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
- Abel theorem, Encyclopedia of Mathematics (standard reference, not scraped)
- MIT 18.100C, Lecture 11: Power Series (standard reference, not scraped)
- S. Semmes, Rice Math 322 notes (standard reference, not scraped)