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The improper -test for rational exponents
Statement
For every rational , and When they converge, their values are respectively and .
Facts & Assumptions
Given: A rational exponent .
The compact-truncation formula and the dyadic bounds are in Truncated integrals of rational powers.
For rational , monotonicity and the rational-power laws give and as , and and as : in the finite-limit cases choose the thresholds and , and use the analogous threshold for divergence (Monotonicity of and of , Laws of rational exponents, Limits at and , and infinite limits at a point, The left and right limits of at , as limits of the restrictions of to and ).
Nonnegative comparison transfers improper convergence (Comparison tests for improper integrals).
Proof
If , then , so [L1] gives .
If , the integral at infinity diverges: for the formula in [L1] is unbounded by [L2], for use the first dyadic bound, and for compare with the constant one.
If , then , so [L1] and from [L2] give convergence at zero with value .
If , the zero-endpoint integral diverges: use the unbounded formula and [L2] for , and the second dyadic bound for .
The alternatives in steps 1.1–1.4 exhaust all rational exponents and establish both thresholds and values.
Depends on
- Truncated integrals of rational powers
- Improper integrals over unbounded intervals
- Improper integrals at a finite singular endpoint
- Comparison tests for improper integrals
- Rational powers $a^r$ of a positive base
- Monotonicity of $r \mapsto a^{r}$ and of $a \mapsto a^{r}$
- Laws of rational exponents
- Limits at $+\infty$ and $-\infty$, and infinite limits at a point
- The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$
Used by
- |x-c|^-1/2 has a convergent improper integral across an interior singularity Example
- ∫_-∞^∞(1+x²)⁻¹ dx converges absolutely Example
- ∫₀¹ x^-1/2 dx=2 Example
- 1/x on [-1,1] has principal value 0 but no improper integral Example
- A rational-kernel Frullani integral Example
- Convergence range of x⁻ᵖ(1+x)^-q on (0,∞) for rational exponents Example
- Gabriel's horn has finite improper volume π Example
- The rational p-threshold reverses between zero and infinity Example
- The scaled Gaussian integral and its parameter derivative Example
- The substitution x=1/t exchanges the two rational p-tests Example
- FALSE: denominator degree one more than numerator degree already forces convergence False statement
- FALSE: finite volume implies finite lateral surface area False statement
- The improper integral of e^-x² over ℝ is finite and positive Lemma
- Conventions and proved scope for improper integrals Remark
- Keyhole contours evaluate Mellin-type rational integrals Theorem
- Rational improper integrals without real poles are upper-half-plane residue sums Theorem
- Separate improper convergence implies convergence of the principal value Theorem
Dependency tree · two levels
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Sources
- William F. Trench, Introduction to Real Analysis, Examples 3.4.1–3 (standard reference, not scraped)