Abel's theorem
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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.
For Abel's theorem on algebraic curves, see Jacobian variety.
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Theorem
Let a = {ai: i ≥ 0} be any real-valued sequence with ai ≥ 0 for all i, and let
- <math>G_a(z) = \sum_{i=0}^{\infty} a_i z^i\,<math>
be the power series with coefficients a. Then,
- <math>\lim_{z\uparrow 1} G_a(z) = \sum_{i=0}^{\infty} a_i,\,<math>
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.
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.
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.
External links
- Abelian theorem (http://planetmath.org/encyclopedia/AbelianTheorem.html) at PlanetMath (http://www.planetmath.org/); a more general look at Abelian theorems of this type.fr:Théorème d'Abel