In mathematics, a large number of methods have been proposed for the summation of divergent series. These generally take the form of some linear functional L with domain contained in some space S of numerical sequences. That is, firstly, a useful method for attributing a sum to a series that doesn't converge should at least be linear. Secondly, the sequence of partial sums of the series is considered, which is an equivalent way of presenting it. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a divergent series is an infinite series that does not converge. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

For any such L, its abelian theorem is the result that if c = (c_n) is a convergent sequence, with limit C, then L(c) = C. An example is given by the Cesàro method, in which L is defined as the limit of the arithmetic means of the first N terms of c, as N tends to infinity. One can prove that if c does converge to C, then so does the sequence (d_N) where For a discussion of convergence and convergent series, see limit (mathematics). ... Limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics, the Cesàro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). ... The word infinity comes from the Latin infinitas or unboundedness. It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ...

d_N = (c₁ + c₂ + ... + c_N)/N.

To see that, subtract C everywhere to reduce to the case C = 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take M large enough to make the initial segment of terms up to c_N average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also. In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. ...

The name derives from Abel's theorem on power series. In that case L is the radial limit (thought of within the complex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with term In real analysis, Abels theorem for power series relates a limit of a power series to the sum of its coefficients. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... A disc of unit radius on a plane is called a unit disc. ...

a_nzⁿ

and set z = r.e^iθ. That theorem has its main interest in the case that the power series has radius of convergence exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] so that the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum of the a_n. When the radius is 1 the power series will have some singularity on |z| = 1; the assertion is that, nonetheless, if the sum of the a_n exists, it is equal to the limit over r. This therefore fits exactly into the abstract picture. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

Partial converses to abelian theorems are called Tauberian theorems. The original result of Alfred Tauber stated that if we assume also Alfred Tauber (1866-1942) was a mathematician. ...

a_n = o(1/n)

(see Big O notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent. This was strengthened by J.E. Littlewood: we need only assume O(1/n). It has been suggested that Landau notation be merged into this article or section. ... John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ...

In the abstract setting, therefore, an abelian theorem states that the domain of L contains convergent sequences, and its values there are equal to the Lim functional's. A tauberian theorem states, under some growth condition, that the domain of L is exactly the convergent sequences and no more.

If one thinks of L as some generalised type of weighted average, taken to the limit, a tauberian theorem allows one to discard the weighting, under the correct hypotheses. There are many applications of this kind of result in number theory, in particular in handling Dirichlet series. Number theory is the formal study of numbers. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ...

The development of the field of tauberian theorems received a fresh turn with Norbert Wiener's very general results, namely Wiener's tauberian theorem and its large collection of corollaries. The central theorem can now be proved by Banach algebra methods, and contains much, though not all, of the previous theory. Norbert Wiener Norbert Wiener (November 26, 1894 - March 18, 1964) was a U.S. mathematician and applied mathematician, especially in the field of electronics engineering. ... In mathematics, Wieners tauberian theorem is a 1932 result of Norbert Wiener. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...