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While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.
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.
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 calculus, the Abel–Dini–Pringsheim theorem is a convergence test which constructs from a divergent series a series that diverges more slowly, and from convergent series one that converges more slowly. [1]: §IX.39 Consequently, for every convergence test based on a particular series there is a series about which the test is inconclusive.
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analyse 1821.
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.