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Improper integrals over unbounded intervals

Statement

Let $f$ be Riemann integrable on every compact interval $[a, R]$ with $R > a$ . The improper integral over the right-unbounded interval is $\int_{a}^{\infty} f (x) d x : = \lim_{R \to \infty} \int_{a}^{R} f (x) d x,$ provided this limit exists as a finite real number. In that case the improper integral converges; otherwise it diverges. In particular, a limit of $+ \infty$ or $- \infty$ is divergence, not convergence to an extended-real value.

If $f$ is Riemann integrable on every $[R, b]$ with $R < b$ , define $\int_{- \infty}^{b} f (x) d x : = \lim_{R \to - \infty} \int_{R}^{b} f (x) d x$ under the same finite-limit convention. All finite integrals use the oriented convention.

Depends on

Dependency tree · 2 levels

Sources

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