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Kummer's Transformation Of Series
In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to series acceleration, accelerate the convergence of an infinite series. The method was first suggested by Ernst Kummer in 1837. Technique Let :A=\sum_^\infty a_n be an infinite sum whose value we wish to compute, and let :B=\sum_^\infty b_n be an infinite sum with comparable terms whose value is known. If the limit :\gamma:=\lim_ \frac exists, then a_n-\gamma \,b_n is always also a sequence going to zero and the series given by the difference, \sum_^\infty (a_n-\gamma\, b_n), converges. If \gamma\neq 0, this new series differs from the original \sum_^\infty a_n and, under broad conditions, converges more rapidly.Holy et al.''On Faster Convergent Infinite Series'' Mathematica Slovaca, January 2008 We may then compute A as :A=\gamma\,B + \sum_^\infty (a_n-\gamma\,b_n), where \gamma B is a constant. Where a_n\neq 0, the terms can be written as the product (1-\gamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Acceleration
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 obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities. Definition Given a sequence :S=\_ having a limit :\lim_ s_n = \ell, an accelerated series is a second sequence :S'=\_ which converges faster to \ell than the original sequence, in the sense that :\lim_ \frac = 0. If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit \ell. The mappings from the original to the transformed series may be linear (as defined in the article sequence transfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. Life Kummer was born in Sorau, Brandenburg (then part of Prussia). He was awarded a PhD from the University of Halle in 1831 for writing a prize-winning mathematical essay (''De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis''), which was eventually published a year later. In 1840, Kummer married Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of Felix Mendelssohn and his sister Rebecca Mendelssohn Bartholdy, the wife of the mathematician Peter Gustav Lejeune Dirichlet. His second wife (whom he married soon after the death of Ottilie in 1848), Bertha Cauer, was a maternal co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Formula For π
In mathematics, the Leibniz formula for , named after Gottfried Leibniz, states that 1-\frac+\frac-\frac+\frac-\cdots=\frac, an alternating series. It is also called the Madhava–Leibniz series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1676. The series for the inverse tangent function, which is also known as Gregory's series, can be given by: : \arctan x = x - \frac + \frac - \frac + \cdots The Leibniz formula for \frac can be obtained by putting x=1 into this series. It also is the Dirichlet -series of the non-principal Dirichlet character of modulus 4 evaluated at s=1, and, therefore, the value of the Dirichlet beta function. Proofs Proof 1 \begin \frac &= \arctan(1) \\ &= \int_0^1 \frac 1 \, dx \\ pt& = \int_0^1\left(\sum_^n (-1)^k x^+\frac\right) \, dx \\ pt& = \left(\sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telescoping Series
In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums only consists of two terms of (a_n) after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. For example, the series :\sum_^\infty\frac (the series of reciprocals of pronic numbers) simplifies as :\begin \sum_^\infty \frac & = \sum_^\infty \left( \frac - \frac \right) \\ & = \lim_ \sum_^N \left( \frac - \frac \right) \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack = 1. \end An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, ''De dimensione parabolae''. In general Telescoping sums are finite sums in which pairs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. Definition The binomial transform, ''T'', of a sequence, , is the sequence defined by :s_n = \sum_^n (-1)^k a_k. Formally, one may write :s_n = (Ta)_n = \sum_^n T_ a_k for the transformation, where ''T'' is an infinite-dimensional operator with matrix elements ''T''''nk''. The transform is an involution, that is, :TT = 1 or, using index notation, :\sum_^\infty T_T_ = \delta_ where \delta_ is the Kronecker delta. The original series can be regained by :a_n=\sum_^n (-1)^k s_k. The binomial transform of a sequence is just the ''n''th forward differences of the sequence, with odd differences carrying a negative sign, namely: :\begin s_0 &= a_0 \\ s_1 &= - (\Delta a) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
