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The rational $p$ -threshold reverses between zero and infinity

Example

The exponents $2$ and $1 / 2$ display the opposite convergence thresholds: $\int_{1}^{\infty} x^{- 2} d x = 1, \int_{0}^{1} x^{- 2} d x diverges,$ while $\int_{0}^{1} x^{- 1 / 2} d x = 2, \int_{1}^{\infty} x^{- 1 / 2} d x diverges .$

Facts & Assumptions

Given: The two rational exponents $2$ and $1 / 2$ .

[L1]

At infinity the $p$ -integral converges exactly for $p > 1$ , while at zero it converges exactly for $p < 1$ (The improper $p$ -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 $p = 2$ proves convergence only at infinity, and [L2] gives the value $1 / (2 - 1) = 1$ there.

L1L2

2.1

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

L1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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, Examples 3.4.1–3 (standard reference, not scraped)

The rational pp-threshold reverses between zero and infinity