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where denotes the limit superior (possibly ; if the limit exists it is the same value). If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
The ratio test is a criterion for the convergence of a series where each term is a nonzero real or complex number. It compares the ratio of consecutive terms and their limits, and may be extended to handle cases where the limit is 1 or fails to exist.
Learn how to test infinite series for convergence using Cauchy's criterion, which relies on bounding sums of terms. Find the statement, explanation, proof and generalization of this method for complete metric spaces.
The elbow method is a heuristic approach to choose the optimal number of clusters in data clustering algorithms. It plots the percentage of variance explained by each cluster against the number of clusters and looks for a knee point in the curve.
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 mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f ( n ) {\displaystyle f(n)} of non-negative real numbers , the series ∑ n = 1 ∞ f ( n ) {\textstyle \sum \limits _{n=1}^{\infty }f(n)} converges if and only if the "condensed ...
Learn about different notions of convergence of measures in mathematics, such as setwise, weak and total variation convergence. See definitions, examples and applications in probability theory and functional analysis.
Dirichlet's test is a method of testing for the convergence of a series, named after Peter Gustav Lejeune Dirichlet. It states that if is a sequence of real numbers and a sequence of complex numbers satisfying is monotonic for every positive integer N, then the series converges.