In real analysis, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... 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. ... Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Finnøy. ...

For Abel's theorem on algebraic curves, see Jacobian variety. In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ...

Theorem

Let a = {a_i: i ≥ 0} be any sequence of real or complex numbers and let

be the power series with coefficients a. Suppose that the series converges. Then,

In the special case where all the coefficients ai are real and a_i ≥ 0 for all i, then the above formula ( * ) holds also when the series does not converge. I.e. in that case both sides of the formula equal .

Remark

In a more general version of this theorem, if r is any nonzero real number for which the series converges, then it follows that

provided we interpret the limit in this formula as a one sided limit, from the left if r is positive and from the right if r is negative.

Applications

The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (i.e. z) approaches 1 from below, even in cases where the radius of convergence, R, of the power series is equal to 1 and we cannot be sure whether the limit should be finite or not. 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...

G_a(z) is called the generating function of the sequence a. Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton-Watson processes. In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... This is a page about mathematics. ... In probability theory, the probability-generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. ... The Galton-Watson process is a stochastic process arising from Francis Galtons statistical investigation of the extinction of surnames. ...

Related concepts

Converses to a theorem like Abel's are called Tauberian theorems: there is no exact converse, but results conditional on some hypothesis. The field of divergent series, and their summation methods, contains many theorems of abelian type and of tauberian type. In mathematics, a large number of methods have been proposed for the summation of divergent series. ... In mathematics, a divergent series is a series that does not converge. ...

External links

• Abelian theorem at PlanetMath; a more general look at Abelian theorems of this type.