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Euler Summation
In the mathematics of convergent and 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 mus ..., 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 su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent Series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_1, a_2, a_3, \ldots) defines a series that is denoted :S=a_1 + a_2 + a_3 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = a_1 +a_2 + \cdots + a_n = \sum_^n a_k. A series is convergent (or converges) if and only if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: *: ++++++\cdots = . * Alternating the signs of reciprocals of powers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. Definition The binomial transform, , of a sequence, , is the sequence defined by s_n = \sum_^n (-1)^k \binom a_k. Formally, one may write s_n = (Ta)_n = \sum_^n T_ a_k for the transformation, where is an infinite-dimensional operator with matrix elements . The transform is an involution, that is, TT = 1 or, using index notation, \sum_^\infty T_ T_ = \delta_ where \delta_ is the Kronecker delta. The original series can be regained by a_n=\sum_^n (-1)^k \binom s_k. The binomial transform of a sequence is just the -th forward differences of the sequence, with odd differences carrying a negative sign, namely: \begin s_0 &= a_0 \\ s_1 &= - (\Delta a)_0 = -a_1+a_0 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . Like an .... 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_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel's Summation Formula
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series. Formula Let (a_n)_^\infty be a sequence of real or complex numbers. Define the partial sum function A by :A(t) = \sum_ a_n for any real number t. Fix real numbers x . Then: :\sum_ a_n\phi(n) = A(y)\phi(y) - A(x)\phi(x) - \int_x^y A(u)\phi'(u)\,du. The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions A and \phi. Variations Taking the left endpoint to be -1 gives the formula :\sum_ a_n\phi(n) = A(x)\phi(x) - \int_0^x A(u)\phi'(u)\,du. If the sequence (a_n) is indexed starting at n = 1, then we may formally define a_0 = 0. The previous formula becomes :\sum_ a_n\phi(n) = A(x)\phi(x) - \int_1^x A(u)\phi'(u)\,du. A common way to apply Abel's summation formula is to take the limit of one of these formulas as x \to \infty. The resulting formulas are :\b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel–Plana Formula
In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that \sum_^f\left(a+n\right)= \frac+\int_^f\left(x\right)dx+i\int_^\fracdt For the case a=0 we have :\sum_^\infty f(n)=\frac + \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac \, dt. It holds for functions ''ƒ'' that are holomorphic in the region Re(''z'') ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that , ''ƒ'', is bounded by ''C''/, ''z'', 1+''ε'' in this region for some constants ''C'', ''ε'' > 0, though the formula also holds under much weaker bounds. . An example is provided by the Hurwitz zeta function, :\zeta(s,\alpha)= \sum_^\infty \frac = \frac + \frac 1 + 2\int_0^\infty\frac\frac, which holds for all s \in \mathbb, . Another powerful example is applying the formula to the function e^n^: we obtain \Gamma(x+1)=\operatorname_\left(e^\right)+\theta(x) where \Gamma(x) is the gamma f ... [...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 th ... [...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 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] |
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 Abelian and Tauberian theorem Abelian theorem: If a series is convergent to then it is Lambert summable to . : Suppose that is Lambert summable to |
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] |
Closed-form Expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are ''n''th root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions. The ''closed-form problem'' arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a ''closed-form expression'' of this object; that is, an expression of this object in terms of previous ways of specifying it. Example: roots of polynomials The quadratic ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |