Abel's theorem

In real analysis, Abel's theorem for power series with non-negative coefficients relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.

Theorem

Let a = {a_i: i ≥ 0} be any real-valued sequence with a_i ≥ 0 for all i, and let

$G_{a}(z)=\sum _{i=0}^{\infty }a_{i}z^{i}.$

be the power series with coefficients a. Then,

$\lim _{z\uparrow 1}G_{a}(z)=\sum _{i=0}^{\infty }a_{i},$

whether or not this sum is finite.

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.

Related concepts

There are many so-called 'abelian' concepts and theorems in mathematics, in particular in abstract algebra and analysis.

External links

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