Cesàro Summation

	Cesàro Summation In mathematical analysis, Cesàro summation (also known as the Cesàro mean or Cesàro limit) assigns values to some Series (mathematics), infinite sums that are Divergent series, not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of the sequence of arithmetic means of the first ''n'' partial sums of the series. This special case of a matrix summability method is named for the Italian analyst Ernesto Cesàro (1859–1906). The term ''summation'' can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the Eilenberg–Mazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that the ''sum'' of that series is 1/2. Definition Let (a_n)_^\infty be a sequence, and let :s_k = a_1 + \cdots + a_k= \sum_^k a_n be its th partial sum. The sequence is called Cesàro summable, with Cesàro sum , if, as tends to infinity, the arithmetic mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	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] [Amazon]
	Abelian And Tauberian Theorems In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Cauchy's Limit Theorem Cauchy's limit theorem, named after the French mathematician Augustin-Louis Cauchy, describes a property of converging sequences. It states that for a converging sequence the sequence of the arithmetic means of its first n members converges against the same limit as the original sequence, that is (a_n) with a_n\to a implies (a_1+\cdots+a_n) / n \ \to a.Konrad Knopp: ''Infinite Sequences and Series''. Dover, 1956, pp. 33-36Harro Heuser: ''Lehrbuch der Analysis – Teil 1'', 17th edition, Vieweg + Teubner 2009, ISBN 9783834807779, pp176-179(German) The theorem was found by Cauchy in 1821, subsequently a number of related and generalized results were published, in particular by Otto Stolz (1885) and Ernesto Cesàro (1888). Related results and generalizations If the arithmetic means in Cauchy's limit theorem are replaced by weighted arithmetic means those converge as well. More precisely for sequence (a_n) with a_n\to a and a sequence of positive real numbers (p_n) with \frac \to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Stolz–Cesàro Theorem In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences. Statement of the theorem for the case Let (a_n)_ and (b_n)_ be two sequences of real numbers. Assume that (b_n)_ is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching + \infty , or strictly decreasing and approaching - \infty ) and the following limit exists: : \lim_ \frac=l.\ Then, the limit : \lim_ \frac=l.\ Statement of the theorem for the case Let (a_n)_ and (b_n)_ be two sequences of real numbers. Assume now that (a_n)\to 0 and (b_n)\to 0 while (b_n)_ is strictly decreasing. If : \lim_ \frac=l,\ then : \lim_ \frac=l.\ Proofs Proof of the theorem for the case Case 1: suppose (b_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Silverman–Toeplitz Theorem In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in series summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a linear sequence transformation that preserves the limits of convergent sequences. The linear sequence transformation can be applied to the divergent sequences of partial sums of divergent series to give those series generalized sums. An infinite matrix (a_)_ with complex-valued entries defines a regular matrix summability method if and only if it satisfies all of the following properties: : \begin & \lim_ a_ = 0 \quad j \in \mathbb & & \text \\ pt& \lim_ \sum_^ a_ = 1 & & \text \\ pt& \sup_i \sum_^ \vert a_ \vert m\end = \begin 1 & 0 & 0 & 0 & 0 & \cdots \\ \frac & \frac & 0 & 0 & 0 & \cdots \\ \frac & \frac & \frac & 0 & 0 & \cdots \\ \frac & \frac & \frac & \frac & 0 & \cdots \\ \frac & \frac & \frac & \frac & \frac & \cdots \\ \vdots & \vdots & \vdots & \vdo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Riesz Mean In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean. Definition Given a series \, the Riesz mean of the series is defined by :s^\delta(\lambda) = \sum_ \left(1-\frac\right)^\delta s_n Sometimes, a generalized Riesz mean is defined as :R_n = \frac \sum_^n (\lambda_k-\lambda_)^\delta s_k Here, the \lambda_n are a sequence with \lambda_n\to\infty and with \lambda_/\lambda_n\to 1 as n\to\infty. Other than this, the \lambda_n are taken as arbitrary. Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of s_n = \sum_^n a_k for some sequence \. Typically, a sequence is summable when the limit \lim_ R_n exists, or the limit \lim_s^\delta(\lambda) exists, although the precise summability theorems in question often impose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Ramanujan Summation Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. Summation Since there are no properties of an entire sum, the Ramanujan summation functions as a property of partial sums. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that: :\begin \frac 1 2 f(0) + f(1) + \cdots + f(n - 1) + \frac 1 2 f(n) &= \frac + \sum_^ f(k) = \sum_^ f(k) - \frac \\ &= \int_0^n f(x)\,dx + \sum_^p \frac\left ^(n) - f^(0)\right+ R_p \end Ramanujan wrote this again for different limits of the integral and the corresponding summation for the case in which ''p'' goes to infinity: : \sum_^f(k)=C+\int_a^x f(t)dt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Perron's Formula In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. Statement Let \ be an arithmetic function, and let : g(s)=\sum_^ \frac be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for \Re(s)>\sigma. Then Perron's formula is : A(x) = ' a(n) =\frac\int_^ g(z)\frac \,dz. Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when ''x'' is an integer. The integral is not a convergent Lebesgue integral; it is understood as the Cauchy principal value. The formula requires that ''c'' > 0, ''c'' > σ, and ''x'' > 0. Proof An easy sketch of the proof comes from taking Abel's sum formula : g(s)=\sum_^ \frac=s\int_^ A(x)x^ dx. This is nothing but a Laplace transform under the variable change x = e^t. Inverting it one gets Perron's formula. Examples Be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Lambert Summation In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory. Definition Define the Lambert kernel by L(x)=\log(1/x)\frac with L(1)=1. Note that L(x^n)>0 is decreasing as a function of n when 0. A sum $\sum_^\infty a_n$ is Lambert summable to $A$ if $\lim_\sum_^\infty a_n L(x^n)=A$ , written $\sum_^\infty a_n=A\,\,(\mathrm)$ . Abelian and Tauberian theorem Abelian theorem: If a series is convergent to $A$ then it is Lambert summable to $A$ . Tauberian theorem : Suppose that $\sum_^\infty a_n$ is Lambert summable to $A ...$ [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Hölder Summation In mathematics, Hölder summation is a method for summing divergent series introduced by . Definition Given a series : a_1+a_2+\cdots, define :H^0_n=a_1+a_2+\cdots+a_n :H^_n=\frac If the limit :\lim_H^k_n exists for some ''k'', this is called the Hölder sum, or the (''H'',''k'') sum, of the series. Particularly, since the Cesàro sum of a convergent series always exists, the Hölder sum of a series (that is Hölder summable) can be written in the following form: :\lim_H^k_n See also Cesàro summation In mathematical analysis, Cesàro summation (also known as the Cesàro mean or Cesàro limit) assigns values to some Series (mathematics), infinite sums that are Divergent series, not necessarily convergent in the usual sense. The Cesàro sum ... References * Summability methods {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Fejér's Theorem In mathematics, Fejér's theorem,Leopold FejérUntersuchungen über Fouriersche Reihen ''Mathematische Annalen''vol. 58 1904, 51-69. named after Hungarian mathematician Lipót Fejér, states the following: Explanation of Fejér's Theorem's Explicitly, we can write the Fourier series of ''f'' as f(x)= \sum_^ c_n \, e^where the nth partial sum of the Fourier series of ''f'' may be written as :s_n(f,x)=\sum_^nc_ke^, where the Fourier coefficients c_k are :c_k=\frac\int_^\pi f(t)e^dt. Then, we can define :\sigma_n(f,x)=\frac\sum_^s_k(f,x) = \frac\int_^\pi f(x-t)F_n(t)dt with ''F''''n'' being the ''n''th order Fejér kernel. Then, Fejér's theorem asserts that \lim_ \sigma_n (f, x) = f(x) with uniform convergence. With the convergence written out explicitly, the above statement becomes \forall \epsilon > 0 \, \exist\, n_0 \in \mathbb: n \geq n_0 \implies , f(x) - \sigma_n(f,x), 0 , \int_ F_n (x) \, dx = \frac \int_ \frac \, dx \leq \frac \int_ \frac \, dx This shows tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]
	Euler–Boole Summation Euler–Boole summation is a method for summing alternating series. The concept is named after Leonhard Euler and George Boole George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H .... Boole published this summation method, using Euler's polynomials, but the method itself was likely already known to Euler. Euler's polynomials are defined by The periodic Euler functions modify these by a sign change depending on the parity of the integer part of x: The Euler–Boole formula to sum alternating series is where a,m,n\in\N, a and f^ is the ''k''th derivative. References Series (mathematics) Summability methods {{math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon]

Abelian and Tauberian theorem