Abel's theorem

Remarks

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

The theorem can also be generalized to account for sums which diverge to infinity. If $${\displaystyle \sum _{k=0}^{\mathrm{\infty }}{a}_k=\mathrm{\infty }}$$ then $${\displaystyle \underset{z\to 1^{-}}{lim}G(z)\to \mathrm{\infty }.}$$

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

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

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

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, ${\displaystyle z}$) approaches ${\displaystyle 1}$ from below, even in cases where the radius of convergence, ${\displaystyle R,}$ of the power series is equal to ${\displaystyle 1}$ 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 $${\displaystyle G_a(z)=\frac{\ln (1+z)}{z},0<z<1,}$$ by integrating the uniformly convergent geometric power series term by term on ${\displaystyle [-z,0]}$; thus the series $${\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{k+1}}$$ converges to ${\displaystyle \ln 2}$ by Abel's theorem. Similarly, $${\displaystyle \sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{2k+1}}$$ converges to ${\displaystyle \arctan 1=\frac{\pi }{4}.}$

${\displaystyle G_a(z)}$ is called the generating function of the sequence ${\displaystyle a.}$ 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 ${\displaystyle a_0,}$ we may assume that ${\displaystyle \sum _{k=0}^{\mathrm{\infty }}{a}_k=0.}$ Let ${\displaystyle s_n=\sum _{k=0}^n{a}_k.}$ Then substituting ${\displaystyle a_k=s_k-s_{k-1}}$ and performing a simple manipulation of the series (summation by parts) results in $${\displaystyle G_a(z)=(1-z)\sum _{k=0}^{\mathrm{\infty }}{s}_k{z}^k.}$$

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