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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.
The test was used 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. For a generalization, see Dirichlet's test.
However, this is not the case, and the patch test is neither sufficient nor necessary for convergence. A broader definition of patch test (applicable to any numerical method, including and beyond finite elements) is any test problem having an exact solution that can, in principle, be exactly reproduced by the numerical approximation. Therefore ...
Bayesian inference (/ ˈ b eɪ z i ə n / BAY-zee-ən or / ˈ b eɪ ʒ ən / BAY-zhən) [1] is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available.
A sequence of functions (fₙ) converges uniformly to f when for arbitrary small ε there is an index N such that the graph of fₙ is in the ε-tube around f whenever n ≥ N.
[89] [277] For example, the problem known as the Gambler's ruin is based on a simple random walk, [195] [278] and is an example of a random walk with absorbing barriers. [ 241 ] [ 279 ] Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods, [ 280 ] and then more detailed solutions were presented ...
The dominated convergence theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above. The assumption of convergence almost everywhere can be weakened to require only convergence in measure.
Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as non-topological convergences , that do not arise from any topological space. [ 1 ]