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t. e. In mathematics, the integral test for convergence is a method used to test infinite series of monotonous 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. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862. [1]
Raabe–Duhamel's test. Let { an } be a sequence of positive numbers. Define. If. exists there are three possibilities: if L > 1 the series converges (this includes the case L = ∞) if L < 1 the series diverges. and if L = 1 the test is inconclusive. An alternative formulation of this test is as follows.
t. e. 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 ...
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted. The n th partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if and only if the sequence of its partial sums tends to a limit ...
Series. In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.
exemplifies the possible results of the test: If p ≤ 0, then the term test identifies the series as divergent. If 0 < p ≤ 1, then the term test is inconclusive, but the series is divergent by the integral test for convergence. If 1 < p, then the term test is inconclusive, but the series is convergent, again by the integral test for convergence.
Romberg's method. In numerical analysis, Romberg's method [1] is used to estimate the definite integral by applying Richardson extrapolation [2] repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally ...