Cartesian closed category: Difference between revisions

Revision as of 14:30, 4 October 2013

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Template:No footnotes 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.

Theorem

Let a = {a_k: 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 Stolz angle, that is, a region of the open unit disk where

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

for some M. Without this restriction, the limit may fail to exist.

Note that $G_{a}(z)$ 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 $G_{a}(z)$ is continuous on [0, 1].

Remarks

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_{k}z^{k}\!

in which the limit is taken from below.

The theorem can also be generalized to account for infinite sums. If

\sum _{k=0}^{\infty }a_{k}=\infty \!

then the limit from below $\lim _{z\to 1^{-}}G_{a}(z)$ 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 ${\frac {1}{1+z}}$ . The series is equal to $1-1+1-1+\cdots$ at $z=1$ , but $1/(1+1)=1/2$ .

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 $a_{k}=(-1)^{k}/(k+1)$ , we obtain $G_{a}(z)=\ln(1+z)/z$ for $0<z<1$ , by integrating the uniformly convergent geometric power series term by term on [-z, 0]; thus the series $\sum _{k=0}^{\infty }(-1)^{k}/(k+1)\!$ converges to ln(2) by Abel's theorem. Similarly, $\sum _{k=0}^{\infty }(-1)^{k}/(2k+1)\!$ converges to arctan(1) = $\pi /4$ .

G_a(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.

Outline of proof

After subtracting a constant from $a_{0}\!$ , we may assume that $\sum _{k=0}^{\infty }a_{k}=0\!$ . Let $s_{n}=\sum _{k=0}^{n}a_{k}\!$ . Then substituting $a_{k}=s_{k}-s_{k-1}\!$ and performing a simple manipulation of the series results in

G_{a}(z)=(1-z)\sum _{k=0}^{\infty }s_{k}z^{k}.\!

Given $\epsilon >0\!$ , pick n large enough so that $|s_{k}|<\epsilon \!$ for all $k\geq n\!$ and note that

\left|(1-z)\sum _{k=n}^{\infty }s_{k}z^{k}\right|\leq \epsilon |1-z|\sum _{k=n}^{\infty }|z|^{k}=\epsilon |1-z|{\frac {|z|^{n}}{1-|z|}}<\epsilon M\!

when z lies within the given Stoltz angle. Whenever z is sufficiently close to 1 we have

\left|(1-z)\sum _{k=0}^{n-1}s_{k}z^{k}\right|<\epsilon ,

so that $|G_{a}(z)|<(M+1)\epsilon \!$ 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 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

Template:PlanetMath (a more general look at Abelian theorems of this type)
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+{{about|Abel's theorem on [[power series]]|Abel's theorem on [[algebraic curve]]s|Abel&ndash;Jacobi map|Abel's theorem on the insolubility of the quintic equation|Abel&ndash;Ruffini theorem|Abel's theorem on linear differential equations|Abel's identity|Abel's theorem on irreducible polynomials|Abel's irreducibility theorem}}
+{{no 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]].
+==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]]

Cartesian closed category: Difference between revisions

Revision as of 14:30, 4 October 2013

Contents

Theorem

Remarks

Applications

Outline of proof

Related concepts

See also

Further reading

External links

Navigation menu

Cartesian closed category: Difference between revisions

Revision as of 14:30, 4 October 2013

Theorem

Remarks

Applications

Outline of proof

Related concepts

See also

Further reading

External links

Navigation menu

Search