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be a power series with real coefficients ak with radius of convergence 1. Suppose that the series
converges. Then G(x) is continuous from the left at , i.e.
The same theorem holds for complex power series
provided that within a Stolz sector, that is, a region of the open unit disk where
for some M. Without this restriction, the limit may fail to exist: for example, the power series
converges to 0 at z = 1, but is unbounded near any point of the form eπi/3n, so the value at z = 1 is not the limit as z tends to 1 in the whole open disk.
Note that 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. Abel's theorem allows us to say more, namely that G(z) is continuous on [0, 1].
As an immediate consequence of this theorem, if z is any nonzero complex number for which the series
converges, then it follows that
in which the limit is taken from below.
The theorem can also be generalized to account for sums which diverge to infinity. If
However, if the series is only known to be divergent, the theorem fails; take for example, the power series for
At the series is equal to but
We also remark the theorem holds for radii of convergence other than : let
be a power series with radius of convergence , and suppose the series converges at . Then is continuous from the left at , i.e.
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (i.e. ) approaches 1 from below, even in cases where the radius of convergence, , 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
by integrating the uniformly convergent geometric power series term by term on ; thus the series
converges to by Abel's theorem. Similarly,
is called the generating function of the sequence . 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 , we may assume that . Let . Then substituting and performing a simple manipulation of the series (summation by parts) results in
Given pick n large enough so that for all and note that
when z lies within the given Stolz angle. Whenever z is sufficiently close to 1 we have
so that when z is both sufficiently close to 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.
- 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.
- 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 (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Weisstein, Eric W. "Abel's Convergence Theorem". MathWorld.