In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Abel's test (also known as Abel's criterion) is a method of testing for the
convergence of an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. The test is named after mathematician
Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with
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 ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. Abel's uniform convergence test is a criterion for 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 if, given any arbitrarily ...
of a
series of
functions dependent on
parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
.
Abel's test in real analysis
Suppose the following statements are true:
#
is a convergent series,
# is a monotone sequence, and
# is bounded.
Then
is also convergent.
It is important to understand that this test is mainly pertinent and
useful in the context of non absolutely convergent series
.
For absolutely convergent series, this theorem, albeit true, is almost self evident.
This theorem can be proved directly using
summation by parts.
Abel's test in complex analysis
A closely related convergence test, also known as Abel's test, can often be used to establish the convergence of a
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 ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
on the boundary of its
circle of convergence. Specifically, Abel's test states that if a sequence of ''positive real numbers''
is decreasing monotonically (or at least that for all ''n'' greater than some natural number ''m'', we have
) with
:
then the power series
:
converges everywhere on the closed
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, except when ''z'' = 1. Abel's test cannot be applied when ''z'' = 1, so convergence at that single point must be investigated separately. Notice that Abel's test implies in particular that the radius of convergence is at least 1. It can also be applied to a power series with radius of convergence ''R'' ≠ 1 by a simple change of variables ''ζ'' = ''z''/''R''.
[(Moretti, 1964, p. 91)] Notice that Abel's test is a generalization of the
Leibniz Criterion by taking ''z'' = −1.
Proof of Abel's test: Suppose that ''z'' is a point on the unit circle, ''z'' ≠ 1. For each
, we define
:
By multiplying this function by (1 − ''z''), we obtain
:
The first summand is constant, the second converges uniformly to zero (since by assumption the sequence
converges to zero). It only remains to show that the series converges. We will show this by showing that it even converges absolutely:
where the last sum is a converging telescoping sum. The absolute value vanished because the sequence
is decreasing by assumption.
Hence, the sequence
converges (even uniformly) on the closed unit disc. If
, we may divide by (1 − ''z'') and obtain the result.
Another way to obtain the result is to apply the
Dirichlet's test. Indeed, for
holds
, hence the assumptions of the Dirichlet's test are fulfilled.
Abel's uniform convergence test
Abel's uniform convergence test is a criterion for 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 if, given any arbitrarily ...
of a series of functions or an
improper integration of functions dependent on
parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of
summation by parts.
The test is as follows. Let be a
uniformly bounded sequence of real-valued
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s on a set ''E'' such that ''g''
''n''+1(''x'') ≤ ''g''
''n''(''x'') for all ''x'' ∈ ''E'' and positive integers ''n'', and let be a sequence of real-valued functions such that the series Σ''f''
''n''(''x'') converges uniformly on ''E''. Then Σ''f''
''n''(''x'')''g''
''n''(''x'') converges uniformly on ''E''.
Notes
References
*Gino Moretti, ''Functions of a Complex Variable'', Prentice-Hall, Inc., 1964
*
*
External links
Proof (for real series) at PlanetMath.org
{{Calculus topics
Convergence tests
Articles containing proofs