Abel's theorem

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.

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 a = {a_k: k ≥ 0} be any sequence of real or complex numbers and let

G_a(z) = \sum_{k=0}^{\infty} a_k z^k\!

be the power series with coefficients a. Suppose that the series $\sum_{k=0}^\infty a_k\!$ converges. Then

\lim_{z\rightarrow 1^-} G_a(z) = \sum_{k=0}^{\infty} a_k,\qquad (*)\!

where the variable z is supposed to be real, or, more generally, to lie within any Stolz angle, that is, a region of the open unit disk where

|1-z|\leq M(1-|z|) \,

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

\sum_{n>0}(z^{3^n}-z^{2\times 3^n})/n

converges to 0 at z=1, but is unbounded near any point of the form e^πi/3ⁿ, so the value at z=1 is not the limit as z tends to 1 in the whole open disk.

Note that $G_a(z)$ 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. Abel's theorem allows us to say more, namely that $G_a(z)$ is continuous on [0, 1].

Remarks

As an immediate consequence of this theorem, if z is any nonzero complex number for which the series $\sum_{k=0}^\infty a_k z^k\!$ converges, then it follows that

\lim_{t\to 1^{-}} G_a(tz) = \sum_{k=0}^{\infty} a_kz^k\!

in which the limit is taken from below.

The theorem can also be generalized to account for infinite sums. If

\sum_{k=0}^\infty a_k = \infty\!

then the limit from below $\lim_{z\to 1^{-}} G_a(z)$ will tend to infinity as well. However, if the series is only known to be divergent, the theorem fails; take for example, the power series for $\frac{1}{1+z}$ . The series is equal to $1 - 1 + 1 - 1 + \cdots$ at $z=1$ , but $1/(1+1)=1/2$ .

Applications

The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (i.e. $z$ ) approaches 1 from below, even in cases where the radius of convergence, $R$ , of the power series is equal to 1 and we cannot be sure whether the limit should be finite or not. See e.g. the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when $a_k = (-1)^k/(k+1)$ , we obtain $G_a(z) = \ln(1+z)/z$ for $0 < z < 1$ , by integrating the uniformly convergent geometric power series term by term on $[-z, 0]$ ; thus the series $\sum_{k=0}^\infty (-1)^k/(k+1)\!$ converges to $\ln(2)$ by Abel's theorem. Similarly, $\sum_{k=0}^\infty (-1)^k/(2k+1)\!$ converges to $\arctan(1) = \pi/4$ .

$G_a(z)$ 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

After subtracting a constant from $a_0 \!$ , we may assume that $\sum_{k=0}^\infty a_k=0\!$ . Let $s_n=\sum_{k=0}^n a_k\!$ . Then substituting $a_k=s_k-s_{k-1}\!$ and performing a simple manipulation of the series results in

G_a(z) = (1-z)\sum_{k=0}^{\infty} s_k z^k.\!

Given $\epsilon > 0\!$ , pick n large enough so that $|s_k| < \epsilon\!$ for all $k\ge n\!$ and note that

\left|(1-z)\sum_{k=n}^\infty s_kz^k \right| \le \epsilon |1-z|\sum_{k=n}^\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

\left|(1-z)\sum_{k=0}^{n-1} s_kz^k \right| < \epsilon,

so that $|G_a(z)| < (M+1)\epsilon \!$ when z is both sufficiently close to 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.org. (a more general look at Abelian theorems of this type)
A.A. Zakharov (2001), "Abel summation method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Abel's Convergence Theorem", MathWorld.