Absolute Convergence

Definition: A (improper) integral $\int_{a}^{b} f (x) d x$ is said to be absolutely convergent if the integral of its absolute value converges:

\int_{a}^{b} | f (x) | d x < \infty .

Implication: If an improper integral is absolutely convergent, then it is necessarily convergent. This follows from the Cauchy criterion for improper integrals.

Proof Sketch: For any $ϵ > 0$ , there exists $M$ such that for all $c, d > M$ , $| \int_{c}^{d} | f (x) | d x | < ϵ$ . Hence, for $c, d > M$ , $| \int_{c}^{d} f (x) d x | \leq \int_{c}^{d} | f (x) | d x < ϵ$ , satisfying the Cauchy criterion for convergence of $\int_{a}^{b} f (x) d x$ .