Abel's theorem

From Wikipedia, the free encyclopedia

For Abel's theorem on algebraic curves, see Abel-Jacobi map.
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.

Contents

[edit] Theorem

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

G_a(z) = \sum_{k=0}^{\infty} a_k z^k\!

be the power series with coefficients a. Suppose that the series \sum_{k=0}^\infty a_k\! converges. Then

\lim_{z\rightarrow 1^-} G_a(z) = \sum_{k=0}^{\infty} a_k.\qquad (*)\!

In the special case where all the coefficients ai are real and ak ≥ 0 for all k, then the above formula ( * ) holds also when the series \sum_{i=0}^\infty a_k\! 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_{k=0}^\infty a_k r^k\! converges, then it follows that

\lim_{z\to r} G_a(z) = \sum_{k=0}^{\infty} a_kr^k\!

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. See e.g. the binomial series.

Ga(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