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.
Comparison tests for improper integrals
Statement
Suppose eventually toward a singular end. If the improper integral of converges there, then the integral of converges. If instead eventually and converges, then converges absolutely and hence converges.
The same assertions hold separately at , at , and at either finite singular endpoint.
Facts & Assumptions
Given: The stated eventual pointwise bounds and local Riemann integrability.
Proper integration preserves pointwise order (If on and both are integrable then ; and ).
A nonnegative improper integral converges exactly when its truncation integrals are bounded (A nonnegative improper integral converges iff its truncated integrals are bounded).
Absolute convergence implies convergence (Absolute convergence implies improper convergence).
Finite initial pieces do not affect convergence (Improper convergence is independent of finite truncations and split points).
Proof
Discard the finite portion before the eventual inequality using [L4]. On every remaining compact truncation, [L1] gives . Convergence of bounds the latter truncations, so [L2] gives convergence of .
If , step 1.1 applied to proves absolute convergence; [L3] then proves convergence of . The argument depends only on the direction of truncation and therefore proves every endpoint form.
Depends on
- A nonnegative improper integral converges iff its truncated integrals are bounded
- Absolute convergence implies improper convergence
- Improper convergence is independent of finite truncations and split points
- If $f \le g$ on $[a,b]$ and both are integrable then $\int_a^b f \le \int_a^b g$; and $m(b-a) \le \int_a^b f \le M(b-a)$
- 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 $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$
Used by
- Limit comparison for positive improper integrals Corollary
- ∫_-∞^∞(1+x²)⁻¹ dx converges absolutely Example
- A rational-kernel Frullani integral Example
- Convergence range of x⁻ᵖ(1+x)^-q on (0,∞) for rational exponents Example
- Dirichlet's test for improper integrals Theorem
- The improper p-test for rational exponents Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 94 results over 18 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, Theorem 3.4.6 (standard reference, not scraped)