Comparison test for improper integrals

Statement

Let $f, g \colon [a, \infty) \to \mathbb{R}$ be Riemann integrable on each $[a, b]$, with $0 \le f(x) \le g(x)$ for all $x \ge a$. If $\int_a^\infty g(x) \, dx$ converges, then $\int_a^\infty f(x) \, dx$ converges, and $$\int_a^\infty f(x) \, dx \le \int_a^\infty g(x) \, dx.$$ If $\int_a^\infty f(x) \, dx$ diverges, then $\int_a^\infty g(x) \, dx$ diverges.

Remarks

The function $b \mapsto \int_a^b f$ is increasing, and it is bounded by $\int_a^\infty g$ by monotonicity. A bounded monotone function has a limit. For example, $0 \le e^{-x^2} \le e^{-x}$ for $x \ge 1$. So $\int_1^\infty e^{-x^2} \, dx$ converges. The same test holds for integrals that are improper at a finite endpoint.

Explanations

L1 human

If a larger function has finite area, a smaller nonnegative function also has finite area. This lets us prove convergence without computing the integral.