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Cauchy's convergence test. The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analyse 1821. [1]
An analogous statement for convergence of improper integrals is proven using integration by parts. If the integral of a function f is uniformly bounded over all intervals , and g is a non-negative monotonically decreasing function , then the integral of fg is a convergent improper integral.
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. Integral test. The series can be compared to an integral to establish convergence or divergence.
The Mandelbrot set ( / ˈmændəlbroʊt, - brɒt /) [1] [2] is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not ...
In mathematics, projections onto convex sets ( POCS ), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. [1] The simplest case, when the sets are affine spaces, was analyzed by John von Neumann.
In geometrical optics, vergence describes the curvature of optical wavefronts. [1] Vergence is defined as. where n is the medium's refractive index and r is the distance from the point source to the wavefront. Vergence is measured in units of dioptres (D) which are equivalent to m −1. [1] This describes the vergence in terms of optical power.
Cauchy–Hadamard theorem. In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, [1] but remained relatively unknown until Hadamard rediscovered it. [2]
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 ...