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Antilimit
In mathematics, the antilimit is the equivalent of a Limit (mathematics), limit for a divergent series. The concept not necessarily unique or well-defined, but the general idea is to find a formula for a series and then evaluate it outside its radius of convergence. Common divergent series See also * Abel summation * Cesàro summation * Lindelöf summation * Euler summation * Borel summation * Mittag-Leffler summation * Lambert summation * Euler–Boole summation and Van Wijngaarden transformation can also be used on divergent series Reference * * {{Series (mathematics) Divergent series Summability methods Sequences and series Mathematical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Summability Methods
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent seri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergent Series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Wijngaarden Transformation
In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series. One algorithm to compute Euler's transform runs as follows: Compute a row of partial sums s_ = \sum_^k(-1)^n a_n and form rows of averages between neighbors s_ = \frac2 The first column s_ then contains the partial sums of the Euler transform. Adriaan van Wijngaarden's contribution was to point out that it is better not to carry this procedure through to the very end, but to stop two-thirds of the way. A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Proces Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965) pp. 51-60 If a_0,a_1,\ldots,a_ are available, then s_ is almost always a better approximation to the sum than s_. In many cases the diagonal terms do not converge in one cycle so process of averaging is to be repeated with diagonal terms by bringing them in a row. (For example, this ... [...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 |
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] |
Mittag-Leffler Summation
In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Definition Let :y(z) = \sum_^\infty y_kz^k be a formal power series in ''z''. Define the transform \scriptstyle \mathcal_\alpha y of \scriptstyle y by :\mathcal_\alpha y(t) \equiv \sum_^\infty \fract^k Then the Mittag-Leffler sum of ''y'' is given by :\lim_\mathcal_\alpha y( z) if each sum converges and the limit exists. A closely related summation method, also called Mittag-Leffler summation, is given as follows . Suppose that the Borel transform \mathcal_1 y(z) converges to an analytic function near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the Mittag-Leffler sum of ''y'' is given by :\int_0^\infty e^ \mathcal_\alpha y(t^\alpha z) \, dt When ''α'' = 1 this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Summation
In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation. Definition There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer. Throughout let denote a formal power series :A(z) = \sum_^\infty a_kz^k, and define the Borel transform of to be its equivalent exponential series :\mathcalA(t) \equiv \sum_^\infty \fract^k. Borel's exponential summation method Let denote the partial sum :A_n(z) = \sum_^n a_k z^k. A weak form of Borel's summation method defines the Borel sum of to be : \lim_ e^\sum_^\infty \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Summation
In the mathematics of convergent series, convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σ''a''''n'', if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series. Euler summation can be generalized into a family of methods denoted (E, ''q''), where ''q'' ≥ 0. The (E, 1) sum is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation; for ''q'' > 0 they are incomparable with Abel summation. Definition For some value ''y'' we may define the Euler sum (if it converges for that value of ''y'') corresponding to a particular formal summation as: : _\, \sum_^\infty a_j := \sum_^\infty \frac \sum_^i \binom y^ a_j . If a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindelöf Summation
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Summation
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent seri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
