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In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test .
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
But if the integral diverges, then the series does so as well. In other words, the series a n {\displaystyle {a_{n}}} converges if and only if the integral converges. p -series test
For example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
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 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.
In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. [1]