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Cauchy criterion for improper integrals

Statement

The integral $\int_{a}^{\infty} f$ converges if and only if, for every $ε > 0$ , there is $A > a$ such that $A \leq u < v ⟹ ∣ \int_{u}^{v} f ∣ < ε .$ At a finite right singular endpoint $b$ , replace the condition by $b - δ < u < v < b$ ; at a finite left endpoint use $a < u < v < a + δ$ ; at $- \infty$ use $u < v \leq - A$ . In each case all displayed proper integrals must exist.

Facts & Assumptions

Given: A locally Riemann-integrable $f$ on the relevant one-ended interval.

[L1]

Adjacent proper integrals add (For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ ).

[L2]

A convergent real sequence is Cauchy, and every Cauchy real sequence converges (Every convergent sequence is Cauchy, The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges).

[L3]

The Archimedean property supplies integer truncations beyond every real bound and reciprocal truncations inside every positive neighborhood (Every complete ordered field is Archimedean, For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

Proof

technique · direct

1.1L1

Put $F (R) = \int_{a}^{R} f$ . If $F (R)$ has a finite limit as $R \to \infty$ , then $∣ F (v) - F (u) ∣ < ε$ for all sufficiently large $u, v$ . By [L1], this difference is $\int_{u}^{v} f$ , proving necessity.

1.2L1L2L3

Conversely, the tail condition and [L3] make the sequence $F (n)$ Cauchy, hence convergent to some $L$ by [L2]. Given $ε > 0$ , choose a large integer $n$ for which $∣ F (n) - L ∣ < ε / 2$ and the tail condition is below $ε / 2$ . For every real $R \geq n$ , [L1] gives $F (R) - L = (F (R) - F (n)) + (F (n) - L)$ , so $F (R) \to L$ .

2.1L3∎

For a finite endpoint use the reciprocal sequence $a + 1 / n$ or $b - 1 / n$ furnished by [L3]; the identical Cauchy argument applies. Reversing the real line gives the $- \infty$ form.

Depends on

Dependency tree · 2 levels

Sources

William F. Trench, Introduction to Real Analysis, Section 3.4