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In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to ...
This convergence result is widely applied to prove the convergence of other series as well, whenever those series's terms can be bounded from above by a suitable geometric series; that proof strategy is the basis for the general ratio test for the convergence of infinite series.
Escalator. For the album by Sam Gopal, see Escalator (album). An escalator is a moving staircase which carries people between floors of a building or structure. It consists of a motor -driven chain of individually linked steps on a track which cycle on a pair of tracks which keep the step tread horizontal.
Central–Mid-Levels escalator. The Central–Mid-Levels escalator and walkway system in Hong Kong is the longest outdoor covered escalator system in the world. The system covers over 800 m (2,600 ft) in distance and traverses an elevation of over 135 m (443 ft) from bottom to top. It opened in 1993 to provide an improved link between Central ...
t. e. In mathematics, a series is, roughly speaking, an addition of infinitely many quantities, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating ...
t. e. 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 general, the most common criteria for pointwise convergence of a periodic function f are as follows: If f satisfies a Holder condition, then its Fourier series converges uniformly. If f is of bounded variation, then its Fourier series converges everywhere. If f is continuous and its Fourier coefficients are absolutely summable, then the ...
In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space . The theorem states that each infinite bounded sequence in has a convergent subsequence. [1] An equivalent formulation is that a ...