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{{about|Abel's theorem on [[power series]]|Abel's theorem on [[algebraic curve]]s|Abel–Jacobi map|Abel's theorem on the insolubility of the quintic equation|Abel–Ruffini theorem|Abel's theorem on linear differential equations|Abel's identity|Abel's theorem on irreducible polynomials|Abel's irreducibility theorem}}
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{{no footnotes|date=February 2013}}
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In [[mathematics]], '''Abel's theorem''' for [[power series]] relates a [[limit (mathematics)|limit]] of a power series to the sum of its [[coefficient]]s. It is named after Norwegian mathematician [[Niels Henrik Abel]].
 
==Theorem==
 
Let ''a'' = {''a''<sub>''k''</sub>: ''k'' ≥ 0} be any sequence of real or [[complex number]]s and let
 
:<math>G_a(z) = \sum_{k=0}^{\infty} a_k z^k\!</math>
 
be the power series with coefficients ''a''. Suppose that the series
<math>\sum_{k=0}^\infty a_k\!</math> converges. Then
 
:<math>\lim_{z\rightarrow 1^-} G_a(z) = \sum_{k=0}^{\infty} a_k,\qquad (*)\!</math>
 
where the variable ''z'' is supposed to be real, or, more generally, to lie within any ''Stolz angle'', that is, a region of the open unit disk where
 
: <math> |1-z|\leq M(1-|z|) \, </math>
 
for some&nbsp;''M''.  Without this restriction, the limit may fail to exist.  
 
Note that <math>G_a(z)</math> is continuous on the real closed interval [0, ''t''] for ''t'' < 1, by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that <math>G_a(z)</math> is continuous on [0, 1].
 
==Remarks==
As an immediate consequence of this theorem, if ''z'' is any nonzero complex number for which the series <math>
\sum_{k=0}^\infty a_k z^k\!</math> converges, then it follows that  
 
:<math>\lim_{t\to 1^{-}} G_a(tz) = \sum_{k=0}^{\infty} a_kz^k\!</math>
 
in which the limit is taken [[one-sided limit|from below]].
 
The theorem can also be generalized to account for infinite sums. If
 
:<math>\sum_{k=0}^\infty a_k = \infty\!</math>
 
then the limit from below <math>\lim_{z\to 1^{-}} G_a(z) </math> will tend to infinity as well. However, if the series is only known to
be divergent, the theorem fails; take for example, the power series for <math>\frac{1}{1+z}</math>. The series is equal to <math>1 - 1 + 1 - 1 + \cdots </math> at <math>z=1</math>, but <math>1/(1+1)=1/2</math>.
 
==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]].  Abel's theorem allows us to evaluate many series in closed form.  For example, when <math> a_k = (-1)^k/(k+1)</math>, we obtain <math>G_a(z) = \ln(1+z)/z </math> for <math> 0 < z < 1 </math>, by integrating the uniformly convergent geometric power series term by term on [''-z'', 0]; thus the series <math>\sum_{k=0}^\infty (-1)^k/(k+1)\!</math> converges to ln(2) by Abel's theorem.  Similarly, <math>\sum_{k=0}^\infty (-1)^k/(2k+1)\!</math> converges to arctan(1) = <math> \pi/4 </math>.
 
''G''<sub>''a''</sub>(''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 [[sequence]]s, such as [[probability-generating function]]s.  In particular, it is useful in the theory of [[Galton&ndash;Watson process]]es.
 
==Outline of proof==
 
After subtracting a constant from <math> a_0 \!</math>, we may assume that <math>\sum_{k=0}^\infty a_k=0\!</math>. Let <math>s_n=\sum_{k=0}^n a_k\!</math>. Then substituting <math>a_k=s_k-s_{k-1}\!</math> and performing a simple manipulation of the series results in
 
:<math>G_a(z) = (1-z)\sum_{k=0}^{\infty} s_k z^k.\!</math>
 
Given <math>\epsilon > 0\!</math>, pick ''n'' large enough so that <math>|s_k| < \epsilon\!</math> for all <math>k\ge n\!</math> and note that
 
:<math>\left|(1-z)\sum_{k=n}^\infty s_kz^k \right| \le \epsilon |1-z|\sum_{k=n}^\infty |z|^k = \epsilon|1-z|\frac{|z|^n}{1-|z|} < \epsilon M \!</math>
 
when ''z'' lies within the given Stoltz angle. Whenever ''z'' is sufficiently close to 1 we have
 
:<math>\left|(1-z)\sum_{k=0}^{n-1} s_kz^k \right| < \epsilon, </math>
 
so that <math>|G_a(z)| < (M+1)\epsilon \!</math> when ''z'' is both sufficiently close to 1 and within the Stoltz angle.
 
==Related concepts==
 
Converses to a theorem like Abel's are called [[Tauberian theorem]]s: 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''.
 
==See also==
* [[Summation by parts]]
* [[Abel's summation formula]]
* [[Nachbin resummation]]
 
==Further reading==
 
*{{Cite book|last=Valerian Ahlfors|first=Lars|date=September 1, 1980|title=Complex Analysis|edition=Third|publisher=McGraw Hill Higher Education|pages=41–42|isbn=0-07-085008-9}} - Ahlfors called it ''Abel's limit theorem''.
 
==External links==
* {{PlanetMath | urlname=AbelianTheorem | title=Abel summability | id=3549}} ''(a more general look at Abelian theorems of this type)''
* {{SpringerEOM | urlname=A/a010170 | title=Abel summation method | author=A.A. Zakharov}}
* {{MathWorld | title=Abel's Convergence Theorem | urlname=AbelsConvergenceTheorem}}
 
[[Category:Theorems in real analysis]]
[[Category:Theorems in complex analysis]]
[[Category:Mathematical series]]
[[Category:Niels Henrik Abel]]
[[Category:Summability methods]]

Latest revision as of 13:14, 25 July 2014

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