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In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
Integral; Direct comparison; Limit comparison ... an improper integral is an extension of the notion of a definite integral ... (Apostol 1974, Theorem 10.33): If a ...
An analogous statement for convergence of improper integrals is proven using integration by parts. If the integral of a function f is uniformly bounded over all intervals , and g is a non-negative monotonically decreasing function , then the integral of fg is a convergent improper integral.
In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [4]Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.
In this case, the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour integration, and the Dirichlet kernel. But since the integrand is an even function, the domain of integration can be extended to the negative real number line as well.
In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement [ edit ]
The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals. If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures. for example:
Lebesgue's monotone convergence theorem; Fatou's lemma; Absolutely continuous; Uniform absolute continuity; Total variation; Radon–Nikodym theorem; Fubini's theorem. Double integral; Vitali set, non-measurable set