|
Cesàro Summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean ) assigns values to some infinite sums that are 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 of its first ''n'' partial sums tends to : :\lim_ \f ... [...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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation By Parts
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Statement Suppose \ and \ are two sequences. Then, :\sum_^n f_k(g_-g_k) = \left(f_g_ - f_m g_m\right) - \sum_^n g_(f_- f_). Using the forward difference operator \Delta, it can be stated more succinctly as :\sum_^n f_k\Delta g_k = \left(f_ g_ - f_m g_m\right) - \sum_^ g_\Delta f_k, Summation by parts is an analogue to integration by parts: :\int f\,dg = f g - \int g\,df, or to Abel's summation formula: :\sum_^n f(k)(g_-g_)= \left(f(n)g_ - f(m) g_m\right) - \int_^n g_ f'(t) dt. An alternative statement is :f_n g_n - f_m g_m = \sum_^ f_k\Delta g_k + \sum_^ g_k\Delta f_k + \sum_^ \Delta f_k \Delta g_k which is analogous to the integration by parts formula for semimartingales. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stolz–Cesàro Theorem
In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem 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: s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Silverman–Toeplitz Theorem
In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit. by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive An with -valued entries defines a regular summability method |
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] |
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) \\ &= \int_0^n f(x)\,dx + \sum_^p \frac\left ^(n) - f^(0)\right+ R_p \end Ramanujan wrote it for the case ''p'' going to infinity: : \sum_^x f(k) = C + \int_a^x f(t)\,dt + \frac 1 2 f(x) + \sum_^\infty \fracf^(x) where ''C'' is a constant specific to the series and it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 uniform convergence, 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 summation formula, 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. Invertin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert Summation
In mathematical analysis, Lambert summation is a summability method for a class of divergent series. Definition A series \sum a_n is ''Lambert summable'' to ''A'', written \sum a_n = A \,(\mathrm), if :\lim_ (1-r) \sum_^\infty \frac = A . If a series is convergent to ''A'' then it is Lambert summable to ''A'' (an Abelian theorem). Examples * \sum_^\infty \frac = 0 \,(\mathrm), where μ is the Möbius function. Hence if this series converges at all, it converges to zero. See also * Lambert series * Abel–Plana formula * 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 ... References * * * Mathematical series Summability methods {{Mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder Summation
In mathematics, Hölder summation is a method for summing divergent series introduced by . Definition Given a series (mathematics), 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 References * * Summability methods {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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\sum_^n s_k(f,x) 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 \, \exist\, n_0 \in \mathbb: n \geq n_0 \implies , f(x) - \sigma_n(f,x), 0,\exist \delta > 0: , x-y, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Boole Summation
Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by : \frac=\sum_^\infty E_n(x)\frac. The concept is named after Leonhard Euler and George Boole. The periodic Euler functions are :\widetilde E_n(x+1)=-\widetilde E_n(x)\text \widetilde E_n(x)=E_n(x) \text 0 |
