Abel's theorem

En mathématiques, 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.

Abel's theorem

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This article is about Abel's theorem on power series. For Abel's theorem on algebraic curves, voir Abel–Jacobi map. For Abel's theorem on the insolubility of the quintic equation, voir Abel–Ruffini theorem. For Abel's theorem on linear differential equations, voir Abel's identity. For Abel's theorem on irreducible polynomials, voir Abel's irreducibility theorem. For Abel's formula for summation of a series, using an integral, voir Abel's summation formula.

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Dans mathématiques, Abel's theorem pour power series relates a limit of a power series to the sum of its coefficients.

It is named after Norwegian mathematician Niels Henrik Abel.

Indice

Contenu
Théorème[]
Remarques[]
Applications[]
Aperçu de la preuve[]
Notions connexes[]
Voir également[]
Lectures complémentaires[]
Liens externes[]

Contenu

Théorème[]

Laisse le Taylor series

{style d'affichage G(X)=somme _{k=0}^{infime }un_{k}x^{k}}

be a power series with real coefficients

{style d'affichage a_{k}}

$"a_{k}"$ avec radius of convergence

{style d'affichage 1.}

$"1."$ Suppose that the series

{somme de style d'affichage _{k=0}^{infime }un_{k}}

converge.
Alors

{style d'affichage G(X)}

$"G(X)"$ is continuous from the left at

{

displaystyle x=1,}

$"{displaystyle$ C'est,

{style d'affichage lim _{

The same theorem holds for complex power series

{style d'affichage G(z)=somme _{k=0}^{infime }un_{k}z ^{k},}

provided that

{

displaystyle zto 1}

$"{displaystyle$ entirely within a single Stolz sector, C'est, a region of the open unit disk where

{style d'affichage |1-z|leq M(1-|z|)}

for some fixed finite

{style d'affichage M>1}

$"{displaystyle$ 1}">.

Without this restriction, the limit may fail to exist: par exemple, the power series

{somme de style d'affichage _{n>0}{frac {z ^{3^{n}}-z ^{2

converge vers

{style d'affichage 0}

$"{style$ à

{style d'affichage z = 1,}

$"{style$ but is unbounded near any point of the form

{style d'affichage e^{

pi i/3^{n}},}

$"{style$ so the value at

{style d'affichage z = 1}

$"z=1"$ is not the limit as

{style d'affichage avec}

$"z"$ tends to 1 in the whole open disk.

Notez que

{style d'affichage G(z)}

$"G(z)"$ is continuous on the real closed interval

{style d'affichage [0,t]}

$"[0,t]"$ pour

{style d'affichage t<1,}

$"{style$ 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

{style d'affichage G(z)}

$"G(z)"$ is continuous on

{style d'affichage [0,1].}

$"{style$

Remarques[]

As an immediate consequence of this theorem, si

{style d'affichage avec}

$"z"$ is any nonzero complex number for which the series

{somme de style d'affichage _{k=0}^{infime }un_{k}z ^{k}}

converge, then it follows that

{style d'affichage lim _{

in which the limit is taken from below.

The theorem can also be generalized to account for sums which diverge to infinity.^{[citation requise]} Si

{somme de style d'affichage _{k=0}^{infime }un_{k}=infty }

alors

{style d'affichage lim _{

Cependant, 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: prendre, par exemple, the power series for

{style d'affichage {frac {1}{1+z}}.}

{style d'affichage z = 1}

$"z=1"$ the series is equal to

{

displaystyle 1-1+1-1+cdots ,}

$"{displaystyle$ mais

{style d'affichage {tfrac {1}{1+1}}={tfrac {1}{2}}.}

$"{style$

We also remark the theorem holds for radii of convergence other than

{

displaystyle R=1}

$"R=1"$ : laisser

{style d'affichage G(X)=somme _{k=0}^{infime }un_{k}x^{k}}

be a power series with radius of convergence

{style d'affichage R,}

$"R,"$ and suppose the series converges at

{

displaystyle x=R.}

$"{displaystyle$ Alors

{style d'affichage G(X)}

$"G(X)"$ is continuous from the left at

{

displaystyle x=R,}

$"{displaystyle$ C'est,

{style d'affichage lim _{

Applications[]

The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (C'est,

{style d'affichage avec}

$"z"$ ) approches

{style d'affichage 1}

$"1"$ from below, even in cases where the radius of convergence,

{style d'affichage R,}

$"R,"$ of the power series is equal to

{style d'affichage 1}

$"1"$ and we cannot be sure whether the limit should be finite or not. See for example, la binomial series. Abel's theorem allows us to evaluate many series in closed form. Par exemple, lorsque

{style d'affichage a_{k}={frac {(-1)^{k}}{k+1}},}

on obtient

{style d'affichage G_{un}(z)={frac {dans(1+z)}{z}},

by integrating the uniformly convergent geometric power series term by term on

{style d'affichage [-z,0]}

$"[-z,0]"$

{somme de style d'affichage _{k=0}^{infime }{frac {(-1)^{k}}{k+1}}}

$"{somme$

converge vers

{style d'affichage ln(2)}

$"ln(2)"$ by Abel's theorem. De la même manière,

{somme de style d'affichage _{k=0}^{infime }{frac {(-1)^{k}}{2k+1}}}

converge vers

{

displaystyle arctan(1)={tfrac {pi }{4}}.}

$"{displaystyle$

{style d'affichage G_{un}(z)}

$"G_{un}(z)"$ is called the generating function of the sequence

{

displaystyle a.}

$"a."$ Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative séquences, tel que probability-generating functions. En particulier, it is useful in the theory of Galton–Watson processes.

Aperçu de la preuve[]

After subtracting a constant from

{style d'affichage a_{0},}

$"{style$ we may assume that

{somme de style d'affichage _{k=0}^{infime }un_{k}=0.}

$"{somme$ Laisser

{style d'affichage s_{n}=somme _{k=0}^{n}un_{k}!.}

$"{style$ Then substituting

{style d'affichage a_{k}=s_{k}-s_{k-1}}

$"{style$ and performing a simple manipulation of the series (summation by parts) résulte en

{style d'affichage G_{un}(z)=(1-z)somme _{k=0}^{infime }s_{k}z ^{k}.}

Given

{

displaystyle varepsilon >0,}

$"varepsilon$ 0,"> pick

{displaystyle n}

$"n"$ large enough so that

{style d'affichage |s_{k}|<varepsilon }

$"{style$ pour tous

{

displaystyle kgeq n}

$"{displaystyle$ and note that

{style d'affichage à gauche|(1-z)somme _{

lorsque

{style d'affichage avec}

$"z"$ lies within the given Stolz angle. Whenever

{style d'affichage avec}

$"z"$ is sufficiently close to

{style d'affichage 1}

$"1"$ Nous avons

{style d'affichage à gauche|(1-z)somme _{k=0}^{n-1}s_{k}z ^{k}droit|<varepsilon ,}

pour que

{style d'affichage à gauche|G_{un}(z)droit|<(

M+1)varepsilon }

$"{style$ lorsque

{style d'affichage avec}

$"z"$ is both sufficiently close to

{style d'affichage 1}

$"1"$ and within the Stolz angle.

Notions connexes[]

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 et of tauberian type.

Voir également[]

Abel's summation formula – Integration by parts version of Abel's method for summation by parts
Nachbin resummation
Summation by parts – Theorem to simplify sums of products of sequences

Lectures complémentaires[]

Ahlfors, Lars Valerian (Septembre 1, 1980). Complex Analysis (Third ed.). McGraw Hill Higher Education. pp. 41–42. ISBN 0-07-085008-9. - Ahlfors called it Abel's limit theorem.

Liens externes[]

Abel summability à PlanèteMath. (a more general look at Abelian theorems of this type)
A.A. Zakharov (2001) [1994], "Abel summation method", Encyclopédie des mathématiques, Presse EMS
Weisstein, Eric W. "Abel's Convergence Theorem". MathWorld.

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