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The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. [1] [2] [3] For a generalization, see Dirichlet's test. [4] [5] [6]
if L > 1 the series converges (this includes the case L = ∞) if L < 1 the series diverges; and if L = 1 the test is inconclusive. An alternative formulation of this test is as follows. Let { a n} be a series of real numbers. Then if b > 1 and K (a natural number) exist such that
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. [1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence , limits , continuity , smoothness , differentiability and integrability .
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing.
2 Proof. 3 Example. 4 ... From Calculus to ... 2011, ISBN 9780817682897, pp. 50; Michele Longo and Vincenzo Valori: The Comparison Test: Not Just for Nonnegative Series.
This test can be used with a power series = = ()where the coefficients c n, and the center p are complex numbers and the argument z is a complex variable.. The terms of this series would then be given by a n = c n (z − p) n.
Many authors do not name this test or give it a shorter name. [2] When testing if a series converges or diverges, this test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-Archimedean ultrametric triangle inequality.
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity. There are many types of sequences and modes of convergence , and different proof techniques may be more appropriate than others for proving each type of convergence of each type ...