Abel's theorem

Template:More 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, who proved it in 1826.

Theorem

Let the Taylor series $$G(x)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^k$$ be a power series with real coefficients $a_k$. Suppose that the series $$\sum _{k=0}^{\mathrm{\infty }}{a}_k$$ converges. Then $G(x)$ is continuous from the left at $x=1,$ that is, $$\underset{x\to 1^{-}}{lim}G(x)=\sum _{k=0}^{\mathrm{\infty }}{a}_k.$$

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

The convergence of $\sum _{k=0}^{\mathrm{\infty }}{a}_k$ implies that the radius of convergence of the power series $\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^k$ is at least 1, ensuring convergence for $|z|<1$.

Also, by the uniform limit theorem, $G(z)$ 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 (by the Weierstrass M-test). Abel's theorem allows us to say more, namely that the restriction of $G(z)$ to $[0,1]$ is continuous.

Stolz sector

20 Stolz sectors, for $M$ ranging from 1.01 to 10. The red lines are the tangents to the cone at the right end.

The Stolz sector $|1-z|\le M(1-|z|)$ has explicit equation$$y^2=-\frac{M^4(x^2-1)-2M^2((x-1)x+1)+2\sqrt{M^4(-2M^2(x-1)+2x-1)}+(x-1{)}^2}{(M^2-1{)}^2}$$and is plotted on the right for various values.

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

Remarks

As an immediate consequence of this theorem, if $z$ is any nonzero complex number for which the series $$\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^k$$ converges, then it follows that $$\underset{t\to 1^{-}}{lim}G(tz)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{z}^k$$ in which the limit is taken from below.

The theorem can also be generalized to account for sums which diverge to infinity; see below.Template:Citation needed If $$\sum _{k=0}^{\mathrm{\infty }}{a}_k=\mathrm{\infty }$$ then $$\underset{x\to 1^{-}}{lim}G(x)\to \mathrm{\infty }.$$

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 $$\frac{1}{1+z}.$$

At $z=1$ the series is equal to $1-1+1-1+\cdots ,$ but $\frac{1}{1+1}=\frac{1}{2}.$

We also remark the theorem holds for radii of convergence other than $R=1$: let $$G(x)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^k$$ be a power series with radius of convergence $R>0,$ and suppose the series converges at $x=R.$ Then $G(x)$ is continuous from the left at $x=R,$ that is, $$\underset{x\to R^{-}}{lim}G(x)=G(R).$$

Applications

Abel's theorem allows us to evaluate many series in closed form if we can first prove that they converge. For example, consider the sequence $$a_k=\frac{(-1{)}^k}{k+1}.$$ The sum $\sum _{k=0}^{\mathrm{\infty }}{a}_k$ converges by the alternating series test, so we can apply Abel's theorem to the corresponding generating function $G_a(z)$. We obtain $$G_a(z)=\frac{\ln (1+z)}{z},0<z<1,$$ by integrating the uniformly convergent geometric power series term by term on $[-z,0]$; thus the series $$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{k+1}$$ converges to $\underset{z\to 1^{-}}{lim}{G}_a(z)=\ln 2$ by Abel's theorem. Similarly, $$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{2k+1}$$ converges to $\arctan 1=\frac{\pi }{4}.$

$G_a(z)$ 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

Source:

Let $S=\sum _{k=0}^{\mathrm{\infty }}{a}_k$ and $s_n=\left(\sum _{k=0}^n{a}_k\right)-S.$ Then substituting $a_k=s_k-s_{k-1}$ and performing a simple manipulation of the series (summation by parts) results in $$G_a(z)=S+(1-z)\sum _{k=0}^{\mathrm{\infty }}{s}_k{z}^k.$$

Given $\epsilon >0,$ pick $n$ large enough so that $|s_k|<\epsilon$ for all $k\ge n.$ We have $$|(1-z)\sum _{k=n}^{\mathrm{\infty }}{s}_k{z}^k|\le \epsilon |1-z|\sum _{k=n}^{\mathrm{\infty }}|z{|}^k=\epsilon |1-z|\frac{|z{|}^n}{1-|z|}<\epsilon M$$ when $z$ lies within the given Stolz angle. Whenever $z$ is sufficiently close to $1$ we have $$|(1-z)\sum _{k=0}^{n-1}{s}_k{z}^k|<\epsilon ,$$ so that $|G_a(z)-S|<(M+1)\epsilon$ when $z$ is both sufficiently close to $1$ and within the Stolz angle.

Divergent case

To prove the case for $\sum _{k=0}^{\mathrm{\infty }}{a}_k=+\mathrm{\infty }$, let $x\in (0,1)$ and $s_n=\sum _{k=0}^n{a}_k$. Since $(s_k{)}_k$ diverges to $+\mathrm{\infty }$, we can find, for any $M\in {ℝ}_{+}$, a $N\in \mathbb{N}$ such that $s_k>3M$ for all $k\ge N$. We write, for $n>N$: $$\sum _{k=0}^n{a}_k{x}^k=\left[(1-x)\sum _{k=0}^{N-1}{s}_k{x}^k\right]+\left\{(1-x)\left[\sum _{k=N}^{n-1}{s}_k{x}^k\right]+s_n{x}^n\right\}$$ Since the first term on the right hand side vanishes as $x\to 1$, we can find an $x_1\in (0,1)$ such that it exceeds $-M$ whenever $x_1<x<1$. The second term may be estimated by: $$(1-x)\left[\sum _{k=N}^{n-1}{s}_k{x}^k\right]+s_n{x}^n>3M\left\{(1-x)\left[\sum _{k=N}^{n-1}{x}^k\right]+x^n\right\}=3M{x}^N.$$ Hence, if we let $x_2=(2/3{)}^{1/N}$, then for $x_2<x<1$ this exceeds $2M$. Combining, we get, for any $n>N$ and $max(x_1,x_2)<x<1$: $$\sum _{k=0}^n{a}_k{x}^k>M.$$ This establishes: $$\underset{x\to 1^{-}}{lim}\underset{n\to \mathrm{\infty }}{lim\; inf}\sum _{k=0}^n{a}_k{x}^k=+\mathrm{\infty }$$

In the absence of additional assumptions, the series $\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^k$ might not converge when $|x|<1$, hence the use of the limit inferior.

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.

Generalizations

In the real case, the functions $x^k$ may be replaced by bounded non-negative functions $u_k(x)$ monotonically decreasing with $k$ (i.e., $u_k(x)\ge u_{k+1}(x)$) such that $\underset{x\to 1^{-}}{lim}{u}_k(x)=1$ for all $k$. This follows from Abel's uniform convergence test and applying the uniform limit theorem.

See also

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Further reading

• Template:Cite book - Ahlfors called it Abel's limit theorem.

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External links

• Template:PlanetMath (a more general look at Abelian theorems of this type)
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• Template:MathWorld