How 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.
Absolute and conditional convergence of improper integrals
Statement
An improper integral of is absolutely convergent when the corresponding improper integral of converges. It is conditionally convergent when the integral of converges but the integral of does not.
For an integral with several singular ends, absolute convergence means absolute convergence on every separately defined one-ended piece. Conditional convergence means convergence of every piece and failure of absolute convergence on at least one piece. Thus the terminology never permits cancellation between distinct singular ends.
Depends on
- Improper integrals over unbounded intervals
- Improper integrals at a finite singular endpoint
- Improper integrals with several singular ends
- If $f,g$ are integrable on $[a,b]$ then so are $\lvert f\rvert$, $f^{2}$, $fg$, $\max(f,g)$ and $\min(f,g)$, and $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$
- Absolute value in an ordered field
Used by
- ∫_-∞^∞(1+x²)⁻¹ dx converges absolutely Example
- A step function whose improper integral is the alternating harmonic series Example
- Conventions and proved scope for improper integrals Remark
- A Dirichlet-type transfer criterion for divergence Theorem
- Absolute convergence implies improper convergence Theorem
- Dirichlet's test for improper integrals Theorem
- Uniform oscillatory tail mass forces failure of absolute convergence Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 70 results over 15 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
- William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)