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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
- The scaled Gaussian integral and its parameter derivative Example
- x⁻ᵃ on (0,1) and on (1,∞) calibrates Lᵖ membership Example
- The improper integral of e^-x² over ℝ is finite and positive Lemma
- Dirichlet's test for improper integrals Theorem
- Euler's Beta integral converges exactly for two positive parameters Theorem
- Euler's Gamma integral converges exactly for positive real parameters Theorem
- The improper p-test for rational exponents Theorem
- The real Gamma function is smooth and its derivatives are logarithmic moments Theorem
Dependency tree · two levels
35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- William F. Trench, Introduction to Real Analysis, Theorem 3.4.6 (standard reference, not scraped)