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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.
The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [ 1 ] Integral test
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. Part of a series of articles about
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 ]
3 Integral calculus. 4 Special functions and numbers. ... Summand limit (term test) Ratio; Root; Integral; Direct comparison; Limit comparison; Alternating series;
Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts. The test is as follows.
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
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.
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