View source for Abel's test

{{Short description|Test for series convergence}}
{{Calculus |Series}}
{{about|the mathematical method of testing for the convergence of an infinite series|the test for determining the flash point of petroleum|Abel test}}
In [[mathematics]], '''Abel's test''' (also known as '''Abel's criterion''') is a method of testing for the [[Convergent series|convergence]] of an [[series (mathematics)|infinite series]]. The test is named after mathematician [[Niels Henrik Abel]], who proved it in 1826.<ref>{{cite journal |first=Niels Henrik |last=Abel |authorlink= Niels Henrik Abel |title=Untersuchungen über die Reihe <math>1+ \frac{m}{x} + \frac{m\cdot (m-1)}{2\cdot 1} x^2 + \frac{m\cdot (m-1)\cdot (m-2)}{3\cdot 2\cdot 1} x^3 + \ldots</math> u.s.w. |journal=[[J. Reine Angew. Math.]] |volume=1 |year=1826 |pages=311–339}}</ref> There are two slightly different versions of Abel's test &ndash; 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 (mathematics)|series]] of [[function (mathematics)|functions]] dependent on [[parameter#Mathematical analysis|parameters]].

== Abel's test in real analysis ==

Suppose the following statements are true:

# <math>\sum a_n </math> is a convergent series,
# <math>b_n </math> is a monotone sequence, and
# <math>b_n </math> is bounded.

Then <math>\sum a_nb_n </math> is also convergent.

It is important to understand that this test is mainly pertinent and 
useful in the context of non absolutely convergent series <math>\sum a_n</math>.
For absolutely convergent series, this theorem, albeit true, is almost self evident.{{citation needed|date=September 2024}}

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]] on the boundary of its [[radius of convergence|circle of convergence]]. Specifically, Abel's test states that if a sequence of ''positive real numbers'' <math>(a_n)</math> is decreasing monotonically (or at least that for all ''n'' greater than some natural number ''m'', we have <math>a_n \geq a_{n+1}</math>) with

:<math>
\lim_{n\rightarrow\infty} a_n = 0
</math>

then the power series

:<math>
f(z) = \sum_{n=0}^\infty a_nz^n
</math>

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''.<ref>(Moretti, 1964, p. 91)</ref> Notice that Abel's test is a generalization of the [[Alternating series test|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 <math>n\geq1</math>, we define

:<math>
f_n(z):=\sum_{k=0}^n a_k z^k.
</math>
By multiplying this function by (1 − ''z''), we obtain
:<math>
\begin{align}
(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{align}
</math>
The first summand is constant, the second converges uniformly to zero (since by assumption the sequence <math>(a_n)</math> converges to zero). It only remains to show that the series converges. We will show this by showing that it even converges absolutely:
<math>
\sum_{k=1}^\infty \left|(a_k - a_{k-1}) z^k \right| = \sum_{k=1}^\infty |a_k-a_{k-1}|\cdot |z|^k \leq \sum_{k=1}^\infty (a_{k-1}-a_{k})
</math>
where the last sum is a converging telescoping sum. The absolute value vanished because the sequence <math>(a_n)</math> is decreasing by assumption.

Hence, the sequence <math>(1-z)f_n(z)</math> converges (even uniformly) on the closed unit disc. If <math>z\not = 1</math>, 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 <math>z\ne 1,\ |z|=1</math> holds <math>\left|\sum_{k=0}^n z^k\right|=\left|\frac{z^{n+1}-1}{z-1}\right|\le \frac{2}{|z-1|}</math>, 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 integral|improper integration]] of functions dependent on [[parameter#Mathematical analysis|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''<sub>''n''</sub>} be a [[uniformly bounded]] sequence of real-valued [[continuous function]]s on a set ''E'' such that ''g''<sub>''n''+1</sub>(''x'')&nbsp;≤&nbsp;''g''<sub>''n''</sub>(''x'') for all ''x''&nbsp;∈&nbsp;''E'' and positive integers ''n'', and let {''f''<sub>''n''</sub>} be a sequence of real-valued functions such that the series Σ''f''<sub>''n''</sub>(''x'') converges uniformly on ''E''.  Then Σ''f''<sub>''n''</sub>(''x'')''g''<sub>''n''</sub>(''x'') converges uniformly on ''E''.

== Notes ==
<references/>

== References ==
*Gino Moretti, ''Functions of a Complex Variable'', Prentice-Hall, Inc., 1964
*{{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | title=Mathematical analysis | publisher=[[Addison-Wesley]] | edition=2nd | isbn=978-0-201-00288-1 | year=1974}}
*{{mathworld|title=Abel's uniform convergence test|urlname=AbelsUniformConvergenceTest}}

== External links ==
* [https://web.archive.org/web/20070312202311/http://planetmath.org/encyclopedia/ProofOfAbelsTestForConvergence.html Proof (for real series) at PlanetMath.org]

{{Calculus topics}}

[[Category:Convergence tests]]
[[Category:Articles containing proofs]]