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In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .
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.
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 mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel, who proved it in 1826. [1] There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform ...
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.
Integral test for convergence. The integral test applied to the harmonic series. Since the area under the curve y = 1/x for x ∈ [1, ∞) is infinite, the total area of the rectangles must be infinite as well. In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence.
The test works because the space of real numbers and the space of complex numbers (with the metric given by the absolute value) are both complete. From here, the series is convergent if and only if the partial sums are a Cauchy sequence. Cauchy's convergence test can only be used in complete metric spaces (such as and ), which are spaces where all Cauchy sequences converge. This is because we ...
Calculus. In mathematics, the ratio test is a test (or "criterion") for the convergence of a series. where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.