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Inmathematics,Abel's test (also known asAbel's criterion) is a method of testing for theconvergence of aninfinite series. The test is named after mathematicianNiels Henrik Abel, who proved it in 1826.^[1] There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used withpower series incomplex analysis.Abel's uniform convergence test is a criterion for theuniform convergence of aseries offunctions dependent onparameters.

Suppose the following statements are true:

1. ${\displaystyle \sum a_n}$ is a convergent series,
2. ${\displaystyle b_n}$ is a monotone sequence, and
3. ${\displaystyle b_n}$ is bounded.

Then${\displaystyle \sum a_n{b}_n}$ is also convergent.

It is important to understand that this test is mainly pertinent and useful in the context of non absolutely convergent series${\displaystyle \sum a_n}$.For absolutely convergent series, this theorem, albeit true, is almost self evident.^{[citation needed]}

This theorem can be proved directly usingsummation by parts.

A closely related convergence test, also known asAbel's test, can often be used to establish the convergence of apower series on the boundary of itscircle of convergence. Specifically, Abel's test states that if a sequence ofpositive real numbers${\displaystyle (a_n)}$ is decreasing monotonically (or at least that for alln greater than some natural numberm, 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 closedunit circle, except whenz = 1. Abel's test cannot be applied whenz = 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 convergenceR ≠ 1 by a simple change of variablesζ =z/R.^[2] Notice that Abel's test is a generalization of theLeibniz Criterion by takingz = −1.

Proof of Abel's test: Suppose thatz 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

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 theDirichlet'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 is a criterion for theuniform convergence of a series of functions or animproper integration of functions dependent onparameters. 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 ofsummation by parts.

The test is as follows. Let {g_n} be auniformly bounded sequence of real-valuedcontinuous functions on a setE such thatg_n+1(x) ≤ g_n(x) for allx ∈ E and positive integersn, and let {f_n} be a sequence of real-valued functions such that the series Σf_n(x) converges uniformly onE. Then Σf_n(x)g_n(x) converges uniformly onE.

• Gino Moretti,Functions of a Complex Variable, Prentice-Hall, Inc., 1964
• Apostol, Tom M. (1974),Mathematical analysis (2nd ed.),Addison-Wesley,ISBN 978-0-201-00288-1
• Weisstein, Eric W."Abel's uniform convergence test".MathWorld.

• Proof (for real series) at PlanetMath.org

• Convergence tests

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