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, who proved it in 1826.^[1]

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

The same theorem holds for complex power series $${\displaystyle G(z)=\sum _{k=0}^{\infty }a_{k}z^{k},}$$ provided that ${\displaystyle z\to 1}$ 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 ${\displaystyle M>1}$. Without this restriction, the limit may fail to exist: for example, the power series $${\displaystyle \sum _{n>0}{\frac {z^{3^{n}}-z^{2\cdot 3^{n}}}{n}}}$$ converges to ${\displaystyle 0}$ at ${\displaystyle z=1,}$ but is unbounded near any other point of the form ${\displaystyle e^{\pi i/3^{n}},}$ so the value at ${\displaystyle z=1}$ is not the limit as ${\displaystyle z}$ tends to 1 in the whole open disk.

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

Also note that by the uniform limit theorem, ${\displaystyle G(z)}$ is continuous on the real closed interval ${\displaystyle [0,t]}$ for ${\displaystyle t<1,}$ 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 ${\displaystyle G(z)}$ to ${\displaystyle [0,1]}$ is continuous.

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

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

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

The theorem can also be generalized to account for sums which diverge to infinity; see below. If $${\displaystyle \sum _{k=0}^{\infty }a_{k}=\infty }$$ then $${\displaystyle \lim _{x\to 1^{-}}G(x)\to \infty .}$$

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

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

We also remark the theorem holds for radii of convergence other than ${\displaystyle R=1}$: let $${\displaystyle G(x)=\sum _{k=0}^{\infty }a_{k}x^{k}}$$ 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, $${\displaystyle \lim _{x\to R^{-}}G(x)=G(R).}$$

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 $${\displaystyle G_{a}(z)={\frac {\ln(1+z)}{z}},\qquad 0<z<1,}$$ by integrating the uniformly convergent geometric power series term by term on ${\displaystyle [-z,0]}$; thus the series $${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+1}}}$$ converges to ${\displaystyle \ln 2}$ by Abel's theorem. Similarly, $${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}}$$ converges to ${\displaystyle \arctan 1={\tfrac {\pi }{4}}.}$

${\displaystyle G_{a}(z)}$ 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.

Source:^[2]

Let ${\displaystyle S=\sum _{k=0}^{\infty }a_{k}}$ and ${\displaystyle s_{n}=\left({\sum _{k=0}^{n}a_{k}}\right)-S\!.}$ Then substituting ${\displaystyle a_{k}=s_{k}-s_{k-1}}$ and performing a simple manipulation of the series (summation by parts) results in $${\displaystyle G_{a}(z)=S+(1-z)\sum _{k=0}^{\infty }s_{k}z^{k}.}$$

Given ${\displaystyle \varepsilon >0,}$ pick ${\displaystyle n}$ large enough so that ${\displaystyle |s_{k}|<\varepsilon }$ for all ${\displaystyle k\geq n}$ 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 ${\displaystyle z}$ lies within the given Stolz angle. Whenever ${\displaystyle z}$ is sufficiently close to ${\displaystyle 1}$ we have $${\displaystyle \left|(1-z)\sum _{k=0}^{n-1}s_{k}z^{k}\right|<\varepsilon ,}$$ so that ${\displaystyle \left|G_{a}(z)-S\right|<(M+1)\varepsilon }$ when ${\displaystyle z}$ is both sufficiently close to ${\displaystyle 1}$ and within the Stolz angle.

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

Note that in the absence of additional assumptions, the series ${\displaystyle \sum _{k=0}^{\infty }a_{k}x^{k}}$ might not converge when ${\displaystyle |x|<1}$, hence the use of the limit inferior.

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.

In the real case, the functions ${\displaystyle x^{k}}$ may be replaced by bounded non-negative functions ${\displaystyle u_{k}(x)}$ monotonically decreasing with ${\displaystyle k}$ (i.e., ${\displaystyle u_{k}(x)\geq u_{k+1}(x)}$) such that ${\displaystyle \lim _{x\to 1^{-}}u_{k}(x)=1}$ for all ${\displaystyle k}$^[3]. This follows from Abel's uniform convergence test and applying the uniform limit theorem.

• Abel's summation formula – Integration by parts version of Abel's method for summation by parts
• Nachbin resummation – Theorem bounding the growth rate of analytic functionsPages displaying short descriptions of redirect targets
• Summation by parts – Theorem to simplify sums of products of sequences

• 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.

• Whittaker, E.T.; Watson, G.N. (2021). A Course of Modern Analysis (Fifth ed.). Cambridge University Press.

• 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.