HOME
*





Alternating Series Test
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. 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. 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 an alternating series converges if the following two conditions are met: # , a_n, decreases monotonically, i.e., , a_, \leq, a_n, , and # \lim_ a_n = 0 Alternating series estimation theorem Moreover, let ''L'' denote the sum of the series, then the partial s ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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 i ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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 Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are needed to provide a complete range of responses, the conn ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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 1902. 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 the higher education, K-12, professional, and library markets. It operates in more than 20 countries around the world.(Jun 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 Boston, Massachusetts, and has approximately 5,000 employees worldwide across nearly 38 countries. It was headquartered at its Stamford, Connecticut, office until April 2014.

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 and functional analysis. His books are standard textbooks in universities in many countries. One of his most well-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 Dr. Stewart in t ...
[...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 and ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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 (*1946) und Wolfgang Spohn (*1950), and a son Ingolf Knopp (1915 – 2008), with the grandchildren Brigitte Knopp (*1952) and Werner Knopp (*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 remained for hi ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  




Dirichlet's Test
In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the ''Journal de Mathématiques Pures et Appliquées'' in 1862. Statement The test states that if \ is a sequence of real numbers and \ a sequence of complex numbers satisfying * \ is monotonically decreasing
heorem 1: Let an ≥ 0 be a decreasing sequence * \lim_a_n = 0 * \left, \sum^_b_n\\leq M for every positive integer ''N'' where ''M'' is some constant, then the series \sum^\infty_a_n b_n converges.


Proof

Let S_n = \sum_^n a_k b_k and B_n = \sum_^n b_k. From

Alternating Series
In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. Examples The geometric series 1/2 − 1/4 %2B 1/8 − 1/16 %2B %E2%8B%AF sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not. The Mercator series provides an analytic expression of the natural logarithm: \sum_^\infty \frac x^n \;=\; \ln (1+x). The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact, \sin x = \sum_^\infty (-1)^n \frac, and \cos x = \sum_^\infty (-1)^n \frac . When the alternating factor is removed from these series one obtains the hyperbolic ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Alternating Series
In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. Examples The geometric series 1/2 − 1/4 %2B 1/8 − 1/16 %2B %E2%8B%AF sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not. The Mercator series provides an analytic expression of the natural logarithm: \sum_^\infty \frac x^n \;=\; \ln (1+x). The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact, \sin x = \sum_^\infty (-1)^n \frac, and \cos x = \sum_^\infty (-1)^n \frac . When the alternating factor is removed from these series one obtains the hyperbolic ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


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. Statement A series :\sum_^\infty a_i is convergent if and only if for every \varepsilon>0 there is a natural number ''N'' such that :, a_+a_+\cdots+a_, 0 there is a number ''N'', such that m ≥ n ≥ N imply :, s_m-s_n, =\left, \sum_^m a_k\<\varepsilon Probably the most interesting part of
his theorem His or HIS may refer to: Computing * Hightech Information System, a Hong Kong graphics card company * Honeywell Information Systems * Hybrid intelligent system * Microsoft Host Integration Server Education * Hangzhou International School, i ...

[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]