Abel's test

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 ${\displaystyle (a_n)}$ is decreasing monotonically (or at least that for all n greater than some natural number m, we have ${\displaystyle a_n\ge a_{n+1}}$) with

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$

then the power series

${\displaystyle f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n}$

converges everywhere on the closed 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 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. 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 ${\displaystyle n\ge 1}$, we define

${\displaystyle f_n(z):=\sum _{k=0}^n{a}_k{z}^k.}$

By multiplying this function by (1 − z), we obtain

${\displaystyle \begin{array}{cc}(1-z)f_n(z)& =\sum _{k=0}^n{a}_k(1-z)z^k=\sum _{k=0}^n{a}_k{z}^k-\sum _{k=0}^n{a}_k{z}^{k+1}=a_0+\sum _{k=1}^n{a}_k{z}^k-\sum _{k=1}^{n+1}{a}_{k-1}{z}^k\\ & =a_0-a_n{z}^{n+1}+\sum _{k=1}^n(a_k-a_{k-1})z^k.\end{array}}$

The first summand is constant, the second converges uniformly to zero (since by assumption the sequence ${\displaystyle (a_n)}$ converges to zero). It only remains to show that the series converges. We will show this by showing that it even converges absolutely: ${\displaystyle \sum _{k=1}^{\mathrm{\infty }}\left|(a_k-a_{k-1})z^k\right|=\sum _{k=1}^{\mathrm{\infty }}|a_k-a_{k-1}|\cdot |z{|}^k\le \sum _{k=1}^{\mathrm{\infty }}(a_{k-1}-a_k)}$ where the last sum is a converging telescoping sum. The absolute value vanished because the sequence ${\displaystyle (a_n)}$ is decreasing by assumption.

Hence, the sequence ${\displaystyle (1-z)f_n(z)}$ converges (even uniformly) on the closed unit disc. If ${\displaystyle z\ne 1}$, 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 ${\displaystyle z\ne 1,|z|=1}$ holds ${\displaystyle \left|\sum _{k=0}^n{z}^k\right|=\left|\frac{z^{n+1}-1}{z-1}\right|\le \frac{2}{|z-1|}}$, 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 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 {g_n} be a uniformly bounded sequence of real-valued continuous functions on a set E such that g_n+1(x) ≤ g_n(x) for all x ∈ E and positive integers n, and let {f_n} be a sequence of real-valued functions such that the series Σf_n(x) converges uniformly on E. Then Σf_n(x)g_n(x) converges uniformly on E.