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Alternating Series Test
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. 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. For a generalization, see Dirichlet's test. History Leibniz discussed the criterion in his unpublished ''De quadratura arithmetica'' of 1676 and shared his result with Jakob Hermann in June 1705 and with Johann Bernoulli in October, 1713. It was only formally published in 1993. Formal statement Alternating series test A series of the form \sum_^\infty (-1)^ a_n = a_0-a_1 + a_2 - a_3 + \cdots where either all ''a''''n'' are positive or all ''a''''n'' are negative, is called an alternating series. The alternating series test guarantees that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Agnew's Theorem
Agnew's theorem, proposed by American mathematician Ralph Palmer Agnew, characterizes reorderings of terms of infinite series that preserve convergence for all series. Statement We call a permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ... p: \mathbb \to \mathbb an ''Agnew permutation'' if there exists K \in \mathbb such that any interval that starts with 1 is mapped by to a union of at most intervals, i.e., \exists K \in \mathbb \, : \; \forall n \in \mathbb \;\; \#_(p( ,\,n) \le K\,, where \#_ counts the number of intervals. Agnew's theorem. p is an Agnew permutation \iff for all converging series of real or complex terms \sum_^\infty a_i\,, the series \sum_^\infty a_ converges to the same sum. Corollary 1. p^ (the inverse of p) is an Agnew permutati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Course In Modern Analysis
''A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions'' (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by Edmund T. Whittaker and George N. Watson, first published by Cambridge University Press in 1915. The first edition was Whittaker's alone, but later editions were co-authored with Watson. History Its first, second, third, and the fourth edition were published in 1902, 1915, 1920, and 1927, respectively. Since then, it has continuously been reprinted and is still in print today. A revised, expanded and digitally reset fifth edition, edited by Victor H. Moll, was published in 2021. The book is notable for being the standard reference and textbook for a generation of Cambridge mathematicians including Littlewood and Godfrey H. Hardy. Mary L. Cartwright studied it as preparation for her final hono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cengage
Cengage Group is an American educational content, technology, and services company for higher education, K–12, professional, and library markets. It operates in more than 20 countries around the world.(June 27, 2014Global Publishing Leaders 2014: Cengage publishersweekly.comCompany Info – Wall Street JournalCengage LearningCompany Overview of Cengage Learning, Inc. BloombergBusiness Company information The company is headquartered in , Massachusetts, and has some 5,000 employees worldwide across nearly 38 countries. It was headquartered at its |
James Stewart (mathematician)
James Drewry Stewart, (March 29, 1941December 3, 2014) was a Canadian mathematician, violinist, and professor emeritus of mathematics at McMaster University. Stewart is best known for his series of calculus textbooks used for high school, college, and university-level courses. Career Stewart received his master of science at Stanford University and his doctor of philosophy from the University of Toronto in 1967. He worked for two years as a postdoctoral fellow at the University of London, where his research focused on harmonic analysis, harmonic and functional analysis. His books are standard textbooks in universities in many countries. One of his best-known textbooks is ''Calculus: Early Transcendentals'' (1995), a set of textbooks which is accompanied by websitefor students. Stewart was also a violinist and a former member of the Hamilton Philharmonic Orchestra. Integral House From 2003 to 2009 a house designed by Brigitte Shim and Howard Sutcliffe was constructed for Ste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Konrad Knopp
Konrad Hermann Theodor Knopp (22 July 1882 – 20 April 1957) was a German mathematician who worked on generalized limits and complex functions. Family and education Knopp was born in 1882 in Berlin to Paul Knopp (1845–1904), a businessman in manufacturing, and Helene (1857–1923), née Ostertun, whose own father was a butcher. Paul's hometown of Neustettin, then part of Germany, became Polish territory after the Second World War and is now called Szczecinek. In 1910, Konrad married the painter Gertrud Kressner (1879–1974). They had a daughter Ortrud Knopp (1911–1976), with the grandchildren Willfried Spohn (1944–2012), Herbert Spohn (born 1946) and Wolfgang Spohn (born 1950), and a son Ingolf Knopp (1915–2008), with the grandchildren Brigitte Knopp (born 1952) and Werner Knopp (born 1954). Konrad was primarily educated in Berlin, with a brief sojourn at the University of Lausanne in 1901 for a single semester, before settling at the University of Berlin, where he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series
In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms converge to 0 monotonically, but this condition is not necessary for convergence. Examples The geometric series − + − + ⋯ sums to . The alternating harmonic series has a finite sum but the harmonic series does not. The series 1-\frac+\frac-\ldots=\sum_^\infty\frac converges to \frac, but is not absolutely convergent. The Mercator series provides an analytic power series expression of the natural logarithm, given by \sum_^\infty \frac x^n = \ln (1+x),\;\;\;, x, \le1, x\ne-1. The functions si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Harmonic Series
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, where \ln is the natural logarithm and \gamma\approx0.577 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. Applications of the harmonic series and its partial sums include Divergence of the sum of the reciprocals of the primes, Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are nee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series
In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms converge to 0 monotonically, but this condition is not necessary for convergence. Examples The geometric series − + − + ⋯ sums to . The alternating harmonic series has a finite sum but the harmonic series does not. The series 1-\frac+\frac-\ldots=\sum_^\infty\frac converges to \frac, but is not absolutely convergent. The Mercator series provides an analytic power series expression of the natural logarithm, given by \sum_^\infty \frac x^n = \ln (1+x),\;\;\;, x, \le1, x\ne-1. The functions si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Convergence Theorem
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. In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers a_1 \le a_2 \le a_3 \le ...\le K converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded. For sums of non-negative increasing sequences 0 \le a_ \le a_ \le \cdots , it says that taking the sum and the supremum can be interchanged. In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
