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Abel's theorem

 
Sci-Tech Dictionary: Abel theorem
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(′ä·bəl ′thir·əm)

(mathematics) A theorem stating that if a power series in z converges for z = a, it converges absolutely for |z| < |a|. A theorem stating that if a power series in z converges to f(z) for |z| < 1 and to a for z = 1, then the limit of f(z) as z approaches 1 equals a. A theorem stating that if the three series with nth term an, bn, and cn = a0bn + a1bn-1 + &cellip; + anb0, respectively, converge, then the third series equals the product of the first two series.

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Wikipedia: Abel's theorem
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Home > Library > Miscellaneous > Wikipedia
This article is about Abel's theorem on power series. For Abel's theorem on algebraic curves, see Abel–Jacobi map. For Abel's theorem on the insolubility of the quintic equation, see Abel–Ruffini theorem. For Abel's theorem on linear differential equations, see Abel's identity.

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

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 (*)\!

where the variable z is supposed to be real, or, more generally, to lie within any Stoltz angle, that is, a region of the open unit disk where

|\Im(z)|\leq M(1-\Re(z)) \,

for some M. Dropping this restriction, the equality may fail to hold.

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_{k=0}^\infty a_k\! does not converge. I.e. in that case both sides of the formula equal +∞.

Remark

As an immediate consequence of this theorem, if z is any nonzero complex number for which the series \sum_{k=0}^\infty a_k z^k\! converges, then it follows that

\lim_{t\to 1^{-}} G_a(tz) = \sum_{k=0}^{\infty} a_kz^k\!

in which the limit is taken from the left.

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.

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.

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This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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