Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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The rational pp-threshold reverses between zero and infinity

Example

The exponents 22 and 1/21/2 display the opposite convergence thresholds: ∫1∞x−2dx=1,∫01x−2dx diverges,\int_1^\infty x^{-2}dx=1,\qquad \int_0^1x^{-2}dx\text{ diverges}, while ∫01x−1/2dx=2,∫1∞x−1/2dx diverges.\int_0^1x^{-1/2}dx=2,\qquad \int_1^\infty x^{-1/2}dx\text{ diverges}.

Facts & Assumptions

Given: The two rational exponents 22 and 1/21/2.

[L1]

At infinity the pp-integral converges exactly for p>1p>1, while at zero it converges exactly for p<1p<1 (The improper pp-test for rational exponents).

[L2]

Convergent values follow from the truncated rational-power formula (Truncated integrals of rational powers).

Verification

technique · direct
1.1

Applying [L1] at p=2p=2 proves convergence only at infinity, and [L2] gives the value 1/(2−1)=11/(2-1)=1 there.

L1L2
2.1

Applying [L1] at p=1/2p=1/2 proves convergence only at zero, and [L2] gives 1/(1−1/2)=21/(1-1/2)=2. These are precisely the four assertions displayed above.

L1L2∎

Depends on

Used by

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Sources