Abel's theorem

From Wikipedia, the free encyclopedia

For Abel's theorem on algebraic curves, see Jacobian variety.

For Abel's theorem on the insolubility of the quintic equation with radicals, see Abel-Ruffini theorem.

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.

1 Theorem
2 Remark
3 Applications
4 Related concepts
5 External links

[edit] Theorem

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

$G_a(z) = \sum_{i=0}^{\infty} a_i z^i\,$

be the power series with coefficients a. Suppose that the series $\sum_{i=0}^\infty a_i$ converges. Then,

$\lim_{z\rightarrow 1^-} G_a(z) = \sum_{i=0}^{\infty} a_i.\qquad (*)$

In the special case where all the coefficients a_i are real and a_i ≥ 0 for all i, then the above formula $( * )$ holds also when the series $\sum_{i=0}^\infty a_i$ 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 r is any nonzero real number for which the series $\sum_{i=0}^\infty a_i r^i$ converges, then it follows that

$\lim_{z\to r} G_a(z) = \sum_{i=0}^{\infty} a_ir^i. \, \ \$

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.

[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.

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.