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The improper $p$ -test for rational exponents

Statement

For every rational $p$ , $\int_{1}^{\infty} x^{- p} d x converges exactly when p > 1,$ and $\int_{0}^{1} x^{- p} d x converges exactly when p < 1.$ When they converge, their values are respectively $1 / (p - 1)$ and $1 / (1 - p)$ .

Facts & Assumptions

Given: A rational exponent $p$ .

[L1]

The compact-truncation formula and the dyadic $p = 1$ bounds are in Truncated integrals of rational powers.

[L2]

For rational $s > 0$ , monotonicity and the rational-power laws give $R^{s} \to \infty$ and $R^{- s} \to 0$ as $R \to \infty$ , and $c^{s} \to 0$ and $c^{- s} \to \infty$ as $c ↓ 0$ : in the finite-limit cases choose the thresholds $R > ε^{- 1 / s}$ and $0 < c < ε^{1 / s}$ , and use the analogous threshold for divergence (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 $A \cap (c, \infty)$ ).

[L3]

Nonnegative comparison transfers improper convergence (Comparison tests for improper integrals).

Proof

technique · cases

1.1L1L2assume-case infinityhigh

If $p > 1$ , then $R^{1 - p} \to 0$ , so [L1] gives $\int_{1}^{R} x^{- p} d x \to 1 / (p - 1)$ .

1.2L1L2L3assume-case infinitylow

If $p \leq 1$ , the integral at infinity diverges: for $0 < p < 1$ the formula in [L1] is unbounded by [L2], for $p = 1$ use the first dyadic bound, and for $p \leq 0$ compare $x^{- p} \geq 1$ with the constant one.

1.3L1L2assume-case zerolow

If $p < 1$ , then $1 - p > 0$ , so [L1] and $c^{1 - p} \to 0$ from [L2] give convergence at zero with value $1 / (1 - p)$ .

1.4L1L2assume-case zerohigh

If $p \geq 1$ , the zero-endpoint integral diverges: use the unbounded formula and [L2] for $p > 1$ , and the second dyadic bound for $p = 1$ .

2.1step 1.1step 1.2step 1.3step 1.4cases-exhaustive∎

The alternatives in steps 1.1–1.4 exhaust all rational exponents and establish both thresholds and values.

Depends on

Dependency tree · 2 levels

Sources

William F. Trench, Introduction to Real Analysis, Examples 3.4.1–3

The improper pp-test for rational exponents