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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Abel's theorem for
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
relates a limit of a power series to the sum of its
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s. It is named after Norwegian mathematician
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
, who proved it in 1826.


Theorem

Let the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
G (x) = \sum_^\infty a_k x^k be a power series with real coefficients a_k with radius of convergence 1. Suppose that the series \sum_^\infty a_k converges. Then G(x) is continuous from the left at x = 1, that is, \lim_ G(x) = \sum_^\infty a_k. The same
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
holds for
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
power series G(z) = \sum_^\infty a_k z^k, provided that z \to 1 entirely within a single ''Stolz sector'', that is, a region of the
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
where , 1-z, \leq M(1-, z, ) for some fixed finite M > 1. Without this restriction, the limit may fail to exist: for example, the power series \sum_ \frac n converges to 0 at z = 1, but is unbounded near any point of the form e^, 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 In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
, t/math> for t < 1, by virtue of the
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...
of the series on
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
subsets of the disk of convergence. Abel's theorem allows us to say more, namely that the restriction of G(z) to
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> is continuous.


Stolz sector

The Stolz sector , 1-z, \leq M(1-, z, ) has explicit equationy^2 = -\fracand is plotted on the right for various values. The left end of the sector is x = \frac, and the right end is x=1. On the right end, it becomes a cone with
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
2\theta where \cos\theta = \frac.


Remarks

As an immediate consequence of this theorem, if z is any nonzero complex number for which the series \sum_^\infty a_k z^k converges, then it follows that \lim_ G(tz) = \sum_^\infty a_kz^k in which the limit is taken from below. The theorem can also be generalized to account for sums which diverge to infinity. If \sum_^\infty a_k = \infty then \lim_ G(z) \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 \frac. At z = 1 the series is equal to 1 - 1 + 1 - 1 + \cdots, but \tfrac = \tfrac. We also remark the theorem holds for radii of convergence other than R = 1: let G(x) = \sum_^\infty a_kx^k be a power series with radius of convergence R, and suppose the series converges at x = R. Then G(x) is continuous from the left at x = R, that is, \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, 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 for example, the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when a_k = \frac, we obtain G_a(z) = \frac, \qquad 0 < z < 1, by integrating the uniformly convergent geometric power series term by term on z, 0/math>; thus the series \sum_^\infty \frac converges to \ln 2 by Abel's theorem. Similarly, \sum_^\infty \frac converges to \arctan 1 = \tfrac. G_a(z) is called the
generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
of the sequence a. Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s, such as
probability-generating function In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often ...
s. In particular, it is useful in the theory of
Galton–Watson process The Galton–Watson process, also called the Bienaymé-Galton-Watson process or the Galton-Watson branching process, is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The ...
es.


Outline of proof

After subtracting a constant from a_0, we may assume that \sum_^\infty a_k=0. Let s_n=\sum_^n a_k\!. Then substituting a_k=s_k-s_ and performing a simple manipulation of the series ( summation by parts) results in G_a(z) = (1-z)\sum_^ s_k z^k. Given \varepsilon > 0, pick n large enough so that , s_k, < \varepsilon for all k \geq n and note that \left, (1-z)\sum_^\infty s_kz^k \ \leq \varepsilon , 1-z, \sum_^\infty , z, ^k = \varepsilon, 1-z, \frac < \varepsilon M when z lies within the given Stolz angle. Whenever z is sufficiently close to 1 we have \left, (1-z)\sum_^ s_kz^k \ < \varepsilon, so that \left, G_a(z)\ < (M+1) \varepsilon 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 In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ...
, and their summation methods, contains many theorems ''of abelian type'' and ''of tauberian type''.


See also

* * *


Further reading

* - Ahlfors called it ''Abel's limit theorem''.


References


External links

* ''(a more general look at Abelian theorems of this type)'' * * {{MathWorld , title=Abel's Convergence Theorem , urlname=AbelsConvergenceTheorem Theorems in real analysis Theorems in complex analysis Series (mathematics) Niels Henrik Abel Summability methods