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.

Theorem[]

Let the Taylor series

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

"{displaystyle

be a power series with real coefficients

a k {displaystyle a_{k}}

"a_{k}" with radius of convergence

1. {displaystyle 1.}

"1." Suppose that the series

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

"{displaystyle

converges.
Then

G ( x ) {displaystyle G(x)}

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

x = 1 , {displaystyle x=1,}

"{displaystyle that is,

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

"{displaystyle

The same theorem holds for complex power series

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

"{displaystyle

provided that

z 1 {displaystyle zto 1}

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

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

"{displaystyle

for some fixed finite

1}"> M > 1 {displaystyle M>1}

"{displaystyle1}">.

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

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

"{displaystyle0}{frac {z^{3^{n}}-z^{2cdot 3^{n}}}{n}}}">

converges to

0 {displaystyle 0}

"{displaystyle at

z = 1 , {displaystyle z=1,}

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

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

"{displaystyle so the value at

z = 1 {displaystyle z=1}

"z=1" is not the limit as

z {displaystyle z}

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

Note that

G ( z ) {displaystyle G(z)}

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

[ 0 , t ] {displaystyle [0,t]}

"[0,t]" for

t < 1 , {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

G ( z ) {displaystyle G(z)}

"G(z)" is continuous on

[ 0 , 1 ] . {displaystyle [0,1].}

"{displaystyle

Remarks[]

As an immediate consequence of this theorem, if

z {displaystyle z}

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

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

"{displaystyle

converges, then it follows that

lim t 1 G ( t z ) = k = 0 a k z k {displaystyle lim _{tto 1^{-}}G(tz)=sum _{k=0}^{infty }a_{k}z^{k}}

"{displaystyle

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

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

"{displaystyle

then

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

"{displaystyle

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

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

"{displaystyle

At

z = 1 {displaystyle z=1}

"z=1" the series is equal to

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

"{displaystyle but

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

"{displaystyle

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

R = 1 {displaystyle R=1}

"R=1": let

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

"{displaystyle

be a power series with radius of convergence

R , {displaystyle R,}

"R," and suppose the series converges at

x = R . {displaystyle x=R.}

"{displaystyle Then

G ( x ) {displaystyle G(x)}

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

x = R , {displaystyle x=R,}

"{displaystyle that is,

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

"{displaystyle

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,

z {displaystyle z}

"z") approaches

1 {displaystyle 1}

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

R , {displaystyle R,}

"R," of the power series is equal to

1 {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

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

"{displaystyle

we obtain

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

"{displaystyle

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

[ z , 0 ] {displaystyle [-z,0]}

"[-z,0]"

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

"{displaystyle

converges to

ln ( 2 ) {displaystyle ln(2)}

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

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

"{displaystyle

converges to

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

"{displaystyle

G a ( z ) {displaystyle G_{a}(z)}

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

a . {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

a 0 , {displaystyle a_{0},}

"{displaystyle we may assume that

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

"{displaystyle Let

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

"{displaystyle Then substituting

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

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

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

"{displaystyle

Given

0,}"> ε > 0 , {displaystyle varepsilon >0,}

"varepsilon 0,"> pick

n {displaystyle n}

"n" large enough so that

| s k | < ε {displaystyle |s_{k}|<varepsilon }

"{displaystyle for all

k n {displaystyle kgeq n}

"{displaystyle and note that

| ( 1 z ) k = n s k z k | ε | 1 z | k = n | z | k = ε | 1 z | | z | n 1 | z | < ε M {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}

"{displaystyle

when

z {displaystyle z}

"z" lies within the given Stolz angle. Whenever

z {displaystyle z}

"z" is sufficiently close to

1 {displaystyle 1}

"1" we have

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

"{displaystyle

so that

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

"{displaystyle when

z {displaystyle z}

"z" is both sufficiently close to

1 {displaystyle 1}

"1" and within the Stolz angle.

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.

See also[]

Further reading[]

  • Ahlfors, Lars Valerian (September 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.

External links[]


Si quieres conocer otros artículos parecidos a Abel's theorem puedes visitar la categoría Mathematical series.

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