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If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. 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]
The geometric series is an infinite series derived from a special type of sequence called a geometric progression, which is defined by just two parameters: the initial term and the common ratio . Finite geometric series have a third parameter, the final term's power.
e. 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]
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.
t. e. 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.
Weierstrass M-test. 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 ...
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.
Calculus. 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 ...