Abel's test


In mathematics, Abel's test is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel, who proved it in 1826. 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. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.

Abel's test in real analysis

Suppose the following statements are true:
  1. is a convergent series,
  2. is a monotone sequence, and
  3. is bounded.
Then is also convergent.
The test is mainly pertinent and useful in the context of non-absolutely convergent series. It 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 on the boundary of its circle of convergence. Specifically, Abel's test states that if a sequence of positive real numbers is decreasing monotonically with
then the power series
converges everywhere on the closed unit circle, except possibly at z = 1. Abel's test cannot be applied when z = 1, so convergence at that single point must be investigated separately.
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. Abel's test is a generalization of the alternating series test, which is the special case 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, we obtain
The first summand is constant, the second converges uniformly 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 on the closed unit disc. If, we may divide by 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.
The following strengthening of the test is also valid: one may replace the condition that is decreasing with the condition that it is decreasing for sufficiently large n, that is, that there is some natural number m for which whenever n > m.

Abel's uniform convergence test

Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. 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 functions on a set such that for all and positive integers, and let be a sequence of real-valued functions such that the series converges uniformly on. Then converges uniformly on.