Abel's theorem

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.

Abel's theorem

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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. For Abel's theorem on irreducible polynomials, see Abel's irreducibility theorem. For Abel's formula for summation of a series, using an integral, see Abel's summation formula.

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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.

Índice

Contents
Theorem[]
Remarks[]
Applications[]
Outline of proof[]
Related concepts[]
See also[]
Further reading[]
External links[]

Theorem[]

Let the Taylor series

{displaystyle G(x)=sum _{k=0}^{infty }a_{k}x^{k}}

be a power series with real coefficients

{displaystyle a_{k}}

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

{displaystyle 1.}

$"1."$ Suppose that the series

{displaystyle sum _{k=0}^{infty }a_{k}}

converges.
Then

{displaystyle G(x)}

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

{displaystyle x=1,}

$"{displaystyle$ that is,

{displaystyle lim _{xto 1^{-}}G(x)=sum _{k=0}^{infty }a_{k}.}

The same theorem holds for complex power series

{displaystyle G(z)=sum _{k=0}^{infty }a_{k}z^{k},}

provided that

{displaystyle zto 1}

$"{displaystyle$ entirely within a single Stolz sector, that is, a region of the open unit disk where

{displaystyle |1-z|leq M(1-|z|)}

for some fixed finite

{displaystyle M>1}

$"{displaystyle$ 1}">.

Without this restriction, the limit may fail to exist: for example, the power series

{displaystyle sum _{n>0}{frac {z^{3^{n}}-z^{2cdot 3^{n}}}{n}}}

converges to

{displaystyle 0}

$"{displaystyle$ at

{displaystyle z=1,}

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

{displaystyle e^{pi i/3^{n}},}

$"{displaystyle$ so the value at

{displaystyle z=1}

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

{displaystyle z}

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

Note that

{displaystyle G(z)}

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

{displaystyle [0,t]}

$"[0,t]"$ for

{displaystyle t<1,}

$"{displaystyle$ 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

{displaystyle G(z)}

$"G(z)"$ is continuous on

{displaystyle [0,1].}

$"{displaystyle$

Remarks[]

As an immediate consequence of this theorem, if

{displaystyle z}

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

{displaystyle sum _{k=0}^{infty }a_{k}z^{k}}

converges, then it follows that

{displaystyle lim _{tto 1^{-}}G(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 sums which diverge to infinity.^{[citation needed]} If

{displaystyle sum _{k=0}^{infty }a_{k}=infty }

then

{displaystyle lim _{zto 1^{-}}G(z)to infty .}

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

{displaystyle {frac {1}{1+z}}.}

{displaystyle z=1}

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

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

$"{displaystyle$ but

{displaystyle {tfrac {1}{1+1}}={tfrac {1}{2}}.}

$"{displaystyle$

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

{displaystyle R=1}

$"R=1"$ : let

{displaystyle G(x)=sum _{k=0}^{infty }a_{k}x^{k}}

be a power series with radius of convergence

{displaystyle R,}

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

{displaystyle x=R.}

$"{displaystyle$ Then

{displaystyle G(x)}

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

{displaystyle x=R,}

$"{displaystyle$ that is,

{displaystyle lim _{xto R^{-}}G(x)=G(R).}

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,

{displaystyle z}

$"z"$ ) approaches

{displaystyle 1}

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

{displaystyle R,}

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

{displaystyle 1}

$"1"$ 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

{displaystyle a_{k}={frac {(-1)^{k}}{k+1}},}

we obtain

{displaystyle G_{a}(z)={frac {ln(1+z)}{z}},qquad 0<z<1,}

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

{displaystyle [-z,0]}

$"[-z,0]"$

{displaystyle sum _{k=0}^{infty }{frac {(-1)^{k}}{k+1}}}

$"{displaystyle$

converges to

{displaystyle ln(2)}

$"ln(2)"$ by Abel's theorem. Similarly,

{displaystyle sum _{k=0}^{infty }{frac {(-1)^{k}}{2k+1}}}

converges to

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

$"{displaystyle$

{displaystyle G_{a}(z)}

$"G_{a}(z)"$ 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 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

{displaystyle a_{0},}

$"{displaystyle$ we may assume that

{displaystyle sum _{k=0}^{infty }a_{k}=0.}

$"{displaystyle$ Let

{displaystyle s_{n}=sum _{k=0}^{n}a_{k}!.}

$"{displaystyle$ Then substituting

{displaystyle a_{k}=s_{k}-s_{k-1}}

$"{displaystyle$ and performing a simple manipulation of the series (summation by parts) results in

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

Given

{displaystyle varepsilon >0,}

$"varepsilon$ 0,"> pick

{displaystyle n}

$"n"$ large enough so that

{displaystyle |s_{k}|<varepsilon }

$"{displaystyle$ for all

{displaystyle kgeq n}

$"{displaystyle$ and note that

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

when

{displaystyle z}

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

{displaystyle z}

$"z"$ is sufficiently close to

{displaystyle 1}

$"1"$ we have

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

so that

{displaystyle left|G_{a}(z)right|<(M+1)varepsilon }

$"{displaystyle$ when

{displaystyle z}

$"z"$ is both sufficiently close to

{displaystyle 1}

$"1"$ and within the Stolz 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[]

Abel summability at PlanetMath. (a more general look at Abelian theorems of this type)
A.A. Zakharov (2001) [1994], "Abel summation method", Encyclopedia of Mathematics, EMS Press
Weisstein, Eric W. "Abel's Convergence Theorem". MathWorld.

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Abel's theorem

Abel's theorem

Contents

Theorem[]

Remarks[]

Applications[]

Outline of proof[]

Related concepts[]

See also[]

Further reading[]

External links[]