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converges absolutely
Example
The whole-line integral converges absolutely. No evaluation of its value is needed.
Facts & Assumptions
Given: on .
Continuous functions are properly Riemann integrable on compact intervals (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, A continuous function on is Riemann integrable, by Heine-Cantor and Riemann's criterion).
For , .
The tail integral converges, and comparison transfers convergence (The improper -test for rational exponents, Comparison tests for improper integrals).
Verification
By [L1], the integral over is proper. On , [L2] and [L3] prove convergence. Substitution gives the identical conclusion on .
Since , . Both tails converge separately and the middle piece is proper, so the mixed integral is absolutely convergent by definition.
Depends on
- Improper integrals with several singular ends
- Absolute and conditional convergence of improper integrals
- Comparison tests for improper integrals
- The improper $p$-test for rational exponents
- If $f,g$ are integrable on $[a,b]$ then so are $\lvert f\rvert$, $f^{2}$, $fg$, $\max(f,g)$ and $\min(f,g)$, and $\bigl\lvert\int_a^b f\bigr\rvert \le \int_a^b\lvert f\rvert$
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion
- The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$
Used by
Nothing in the library uses this result yet.
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Sources
- William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)