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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 a n 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.
1.9 Absolute convergence test. 1.10 Alternating series test. 1.11 Dirichlet's test. ... The ratio test may be inconclusive when the limit of the ratio is 1 ...
The same definition can be used for series = whose terms are not numbers but rather elements of an arbitrary abelian topological group.In that case, instead of using the absolute value, the definition requires the group to have a norm, which is a positive real-valued function ‖ ‖: + on an abelian group (written additively, with identity element 0) such that:
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 fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series. Integral test. The series can be ...
In mathematics, convergence tests are methods to determine if an infinite series converges or diverges. Pages in category "Convergence tests" The following 17 pages are in this category, out of 17 total.
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 mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It depends on the quantity | |, where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.