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Series. In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.
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]
e. In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed or with an > 0 for all n. Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit.
Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form = (), if {} is monotonically decreasing, and has a limit of 0 at infinity, then the series converges.
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
v. t. e. 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. [1]
Category. : Convergence tests. In mathematics, convergence tests are methods to determine if an infinite series converges or diverges.
Leibniz theorem (named after Gottfried Wilhelm Leibniz) may refer to one of the following: Product rule in differential calculus. General Leibniz rule, a generalization of the product rule. Leibniz integral rule. The alternating series test, also called Leibniz's rule. The Fundamental theorem of calculus, also called Newton-Leibniz theorem.