Abel's test
Encyclopedia
In mathematics
, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series
. The test is named after mathematician Niels 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 complex analysis
.
1.
is a convergent sequence
2. {
} is a monotone sequence
3. {
} is bounded,
then
is also convergent.
. Specifically, Abel's test states that if
and the series
converges when |z| < 1 and diverges when |z| > 1, and the coefficients {an} are positive real numbers decreasing monotonically toward the limit zero for n > m (for large enough n, in other words), then the power series for f(z) converges everywhere on the unit circle
, 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 can also be applied to a power series with radius of convergence R ≠ 1 by a simple change of variables ζ = z/R.
Proof of Abel's test: Suppose that z is a point on the unit circle, z ≠ 1. Then
so that, for any two positive integers p > q > m, we can write
where Sp and Sq are partial sums:
But now, since |z| = 1 and the an are monotonically decreasing positive real numbers when n > m, we can also write
Now we can apply Cauchy's criterion to conclude that the power series for f(z) converges at the chosen point z ≠ 1, because sin(½θ) ≠ 0 is a fixed quantity, and aq+1 can be made smaller than any given ε > 0 by choosing a large enough q.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
. The test is named after mathematician Niels Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
. 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 complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
.
Abel's test in real analysis
If the following statements are true:1.
is a convergent sequence2. {
} is a monotone sequence3. {
} is bounded, then
is also convergent.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 on the boundary of its circle of convergenceRadius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
. Specifically, Abel's test states that if
and the series
converges when |z| < 1 and diverges when |z| > 1, and the coefficients {an} are positive real numbers decreasing monotonically toward the limit zero for n > m (for large enough n, in other words), then the power series for f(z) converges everywhere on the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
, 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 can also be applied to a power series with radius of convergence R ≠ 1 by a simple change of variables ζ = z/R.
Proof of Abel's test: Suppose that z is a point on the unit circle, z ≠ 1. Then
so that, for any two positive integers p > q > m, we can write
where Sp and Sq are partial sums:
But now, since |z| = 1 and the an are monotonically decreasing positive real numbers when n > m, we can also write
Now we can apply Cauchy's criterion to conclude that the power series for f(z) converges at the chosen point z ≠ 1, because sin(½θ) ≠ 0 is a fixed quantity, and aq+1 can be made smaller than any given ε > 0 by choosing a large enough q.