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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.

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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.

Related concepts

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

External links

  • Abelian theorem (http://planetmath.org/encyclopedia/AbelianTheorem) at PlanetMath (http://www.planetmath.org/); a more general look at Abelian theorems of this type.



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