Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
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FALSE: denominator degree one more than numerator degree already forces convergence

Statement

False claim: if R=p/q is rational and deg⁡q=deg⁡p+1, then ∫−∞∞R(x) dx must converge.

Facts & Assumptions

Given: The rational function R(x)=1/(1+x) on (0,∞), or symmetrically 1/x on the punctured real line.

[L1]

The two-degree gap in the rational residue theorem is the one used for unconditional improper convergence (Rational improper integrals without real poles are upper-half-plane residue sums).

[L2]

The rational p-test diverges at exponent 1 (The improper p-test for rational exponents).

Refutation

technique · direct
1.1

A degree drop by one means that at infinity the rational function behaves like c/x with c≠0. The model case is exactly the p=1 tail in [L2], whose integral diverges logarithmically.

L2
2.1

Therefore the hypothesis deg⁡q=deg⁡p+1 does not force improper [step 1.1, L1] ∎ convergence. The theorem uses the stronger two-degree gap for a reason.

L1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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