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Śleszyński–Pringsheim Theorem
In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century. It states that if a_n, b_n, for n=1,2,3,\ldots are real numbers and , b_n, \geq , a_n, +1 for all n, then : \cfrac converges absolutely to a number f satisfying 0<, f, <1, meaning that the series : where are the convergents of the continued fraction, converges absolutely. See also *Convergence problem
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Convergent (continued Fraction)
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like :a_0 + \cfrac or an infinite continued fraction like :a_0 + \cfrac Typically, such a continued fraction is obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In the ''finite'' case, the iteration/recursion is stopped after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an ''infinite'' continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a_i are called the coefficients or terms of the continued fraction. Simple co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite. Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences \,\ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction". Formulation A continued fraction is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivan Śleszyński
Ivan () is a Slavic male given name, connected with the variant of the Greek name (English: John) from Hebrew meaning 'God is gracious'. It is associated worldwide with Slavic countries. The earliest person known to bear the name was the Bulgarian Saint Ivan of Rila. It is very popular in Russia, Ukraine, Croatia, Serbia, Bosnia and Herzegovina, Slovenia, Bulgaria, Belarus, North Macedonia, and Montenegro and has also become more popular in Romance-speaking countries since the 20th century. Etymology Ivan is the common Slavic Latin spelling, while Cyrillic spelling is two-fold: in Bulgarian, Russian, Macedonian, Serbian and Montenegrin it is , while in Belarusian and Ukrainian it is . The Old Church Slavonic (or Old Cyrillic) spelling is . It is the Slavic relative of the Latin name , corresponding to English '' John''. This Slavic version of the name originates from New Testament Greek (''Iōánnēs'') rather than from the Latin . The Greek name is in turn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Pringsheim
Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts. He was the father-in-law of the author and Nobel Prize winner Thomas Mann. Family and academic career Pringsheim was born in Ohlau, Province of Silesia (now Oława, Poland). He came from an extremely wealthy Silesian merchant family with Jewish roots. He was the first-born child and only son of the Upper Silesian railway entrepreneur and coal mine owner Rudolf Pringsheim (1821–1901) and his wife Paula, née Deutschmann (1827–1909). He had a younger sister, Martha. Pringsheim attended the Maria Magdalena Gymnasium (school), Gymnasium in Breslau, where he excelled in music and mathematics. Starting in 1868 he studied mathematics and physics in Berlin and at the Ruprecht Karl University in Heidelberg. In 1872 he was awarded a doctorate in mathematics, studying under Leo Königsberger. In 1875, he moved from Berlin, where his parents lived, to Munich to earn his habilitati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent (continued Fraction)
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like :a_0 + \cfrac or an infinite continued fraction like :a_0 + \cfrac Typically, such a continued fraction is obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In the ''finite'' case, the iteration/recursion is stopped after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an ''infinite'' continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a_i are called the coefficients or terms of the continued fraction. Simple co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said to converge absolutely if \textstyle\sum_^\infty \left, a_n\ = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x)\,dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty , f(x), dx = L. A convergent series that is not absolutely convergent is called conditionally convergent. Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence Problem
In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ''a''''i'' and partial denominators ''b''''i'' that are sufficient to guarantee the convergence of the infinite continued fraction : x = b_0 + \cfrac.\, This convergence problem is inherently more difficult than the corresponding problem for infinite series. Elementary results When the elements of an infinite continued fraction consist entirely of positive real numbers, the determinant formula can easily be applied to demonstrate when the continued fraction converges. Since the denominators ''B''''n'' cannot be zero in this simple case, the problem boils down to showing that the product of successive denominators ''B''''n''''B''''n''+1 grows more quickly than the product of the partial numerators ''a''1''a''2''a''3...''a''''n''+1. The convergence problem is much more difficult when the elements of the continued fraction are complex numbers. Per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fractions
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite. Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences \,\ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction". Formulation A continued fractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
