Editing Abel's theorem

{{Short description|Power series theorem in mathematics}}
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{{more footnotes|date=February 2013}}
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]], who proved it in 1826.<ref>{{cite journal |last=Abel |first=Niels Henrik |year=1826 |title=Untersuchungen über die Reihe <math>1+ \frac{m}{1} x + \frac{m\cdot (m-1)}{2\cdot 1} x^2 + \frac{m\cdot (m-1)\cdot (m-2)}{3\cdot 2\cdot 1} x^3 + \ldots</math> u.s.w. |journal=[[J. Reine Angew. Math.]] |volume=1 |pages=311–339 |authorlink=Niels Henrik Abel}}</ref>

==Theorem==

Let the [[Taylor series]]
<math display=block>G (x) = \sum_{k=0}^\infty a_k x^k</math>
be a power series with [[real number|real]] coefficients <math>a_k</math> with [[radius of convergence]] <math>1.</math>  Suppose that the series 
<math display=block>\sum_{k=0}^\infty a_k</math> 
[[convergent series|converges]]. 
Then <math>G(x)</math> is [[Continuous function#Directional and semi-continuity|continuous from the left]] at <math>x = 1,</math> that is,
<math display=block>\lim_{x\to 1^-} G(x) = \sum_{k=0}^\infty a_k.</math>

The same [[theorem]] holds for [[complex number|complex]] power series 
<math display=block>G(z) = \sum_{k=0}^\infty a_k z^k,</math> 
provided that <math>z \to 1</math> entirely within a single ''Stolz sector'', that is, a region of the [[open disk|open]] [[unit disk]] where
<math display=block>|1-z| \leq M(1-|z|)</math>
for some fixed finite <math>M > 1</math>.  Without this restriction, the limit may fail to exist: for example, the power series
<math display=block>\sum_{n>0} \frac{z^{3^n}-z^{2\cdot 3^n}} n</math>
converges to <math>0</math> at <math>z = 1,</math> but is [[bounded function|unbounded]] near any point of the form <math>e^{\pi i/3^n},</math> so the value at <math>z = 1</math> is not the limit as <math>z</math> tends to 1 in the whole open disk.

Note that <math>G(z)</math> is continuous on the real [[closed interval]] <math>[0, t]</math> for <math>t < 1,</math> by virtue of the [[uniform convergence]] of the series on [[compact space|compact]] subsets of the disk of convergence. Abel's theorem allows us to say more, namely that the restriction of <math>G(z)</math> to <math>[0, 1]</math> is continuous.

=== Stolz sector ===
[[File:Stolz sector plot.svg|thumb|20 Stolz sectors, for <math>M</math> ranging from 1.01 to 10. The red lines are the tangents to the cone at the right end.]]
The Stolz sector <math>|1-z|\leq M(1-|z|)</math> has explicit equation<math display="block">y^2 = -\frac{M^4 (x^2 - 1) - 2 M^2 ((x - 1) x + 1) + 2 \sqrt{M^4 (-2 M^2 (x - 1) + 2 x - 1)} + (x - 1)^2}{(M^2 - 1)^2}</math>and is plotted on the right for various values.

The left end of the sector is <math>x = \frac{1-M}{1+M}</math>, and the right end is <math>x=1</math>. On the right end, it becomes a cone with [[angle]] <math>2\theta</math> where <math>\cos\theta = \frac{1}{M}</math>.

==Remarks==

As an immediate consequence of this theorem, if <math>z</math> is any nonzero complex number for which the series 
<math display=block>\sum_{k=0}^\infty a_k z^k</math> 
converges, then it follows that
<math display=block>\lim_{t\to 1^{-}} G(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 sums which diverge to infinity.{{citation needed|date=May 2015}} If
<math display=block>\sum_{k=0}^\infty a_k = \infty</math>
then 
<math display=block>\lim_{z\to 1^{-}} G(z) \to \infty.</math>

However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for 
<math display=block>\frac{1}{1+z}.</math>

At <math>z = 1</math> the series is equal to <math>1 - 1 + 1 - 1 + \cdots,</math> but <math>\tfrac{1}{1+1} = \tfrac{1}{2}.</math>

We also remark the theorem holds for radii of convergence other than <math>R = 1</math>: let 
<math display=block>G(x) = \sum_{k=0}^\infty a_kx^k</math> 
be a power series with radius of convergence <math>R,</math> and suppose the series converges at <math>x = R.</math> Then <math>G(x)</math> is continuous from the left at <math>x = R,</math> that is,
<math display=block>\lim_{x\to R^-}G(x) = G(R).</math>

==Applications==

The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, <math>z</math>) approaches <math>1</math> from below, even in cases where the [[radius of convergence]], <math>R,</math> of the power series is equal to <math>1</math> and we cannot be sure whether the limit should be finite or not. See for example, the [[binomial series]]. Abel's theorem allows us to evaluate many series in closed form. For example, when 
<math display=block>a_k = \frac{(-1)^k}{k+1},</math>
we obtain 
<math display=block>G_a(z) = \frac{\ln(1+z)}{z}, \qquad 0 < z < 1,</math> 
by integrating the uniformly convergent geometric power series term by term on <math>[-z, 0]</math>; thus the series 
<math display=block>\sum_{k=0}^\infty \frac{(-1)^k}{k+1}</math> 
converges to <math>\ln 2</math> by Abel's theorem. Similarly, 
<math display=block>\sum_{k=0}^\infty \frac{(-1)^k}{2k+1}</math> 
converges to <math>\arctan 1 = \tfrac{\pi}{4}.</math>

<math>G_a(z)</math> is called the [[generating function]] of the sequence <math>a.</math> 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 ([[summation by parts]]) results in
<math display=block>G_a(z) = (1-z)\sum_{k=0}^{\infty} s_k z^k.</math>

Given <math>\varepsilon > 0,</math> pick <math>n</math> large enough so that <math>|s_k| < \varepsilon</math> for all <math>k \geq n</math> and note that
<math display=block>\left|(1-z)\sum_{k=n}^\infty s_kz^k \right| \leq \varepsilon |1-z|\sum_{k=n}^\infty |z|^k = \varepsilon|1-z|\frac{|z|^n}{1-|z|} < \varepsilon M </math>
when <math>z</math> lies within the given Stolz angle. Whenever <math>z</math> is sufficiently close to <math>1</math> we have
<math display=block>\left|(1-z)\sum_{k=0}^{n-1} s_kz^k \right| < \varepsilon,</math>
so that <math>\left|G_a(z)\right| < (M+1) \varepsilon</math> when <math>z</math> is both sufficiently close to <math>1</math> and within the Stolz angle.

==Related concepts==

[[Converse (logic)|Converse]]s 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''.

==See also==

* {{annotated link|Abel's summation formula}}
* {{annotated link|Nachbin resummation}}
* {{annotated link|Summation by parts}}

==Further reading==

* {{Cite book|last=Ahlfors|first=Lars Valerian|authorlink=Lars Ahlfors|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''.

==References==
{{Reflist}}

==External links==
* {{PlanetMath | urlname=abelsummability | title=Abel summability}} ''(a more general look at Abelian theorems of this type)''
* {{SpringerEOM | 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:Series (mathematics)]]
[[Category:Niels Henrik Abel]]
[[Category:Summability methods]]