Abel's theorem

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In mathematics, 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.

[edit] Theorem

Let a = {a_k: k ≥ 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 a_i are real and a_k ≥ 0 for all k, then the above formula ( * ) holds also when the series does not converge. I.e. in that case both sides of the formula equal +∞.

[edit] Remark

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

in which the limit is taken from the left.

[edit] 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. See e.g. the binomial series.

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.

[edit] 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.

[edit] External links

• Abel summability at PlanetMath. (a more general look at Abelian theorems of this type)
• A.A. Zakharov (2001), "Abel summation method", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104