double series

Theorem.  If the double series

m=1n=1amn=n=1a1n+n=1a2n+n=1a3n+ (1)

converges and if it remains convergentMathworldPlanetmathPlanetmath when the of the partial series are replaced with their absolute valuesMathworldPlanetmathPlanetmathPlanetmath, i.e. if the series

n=1|a1n|+n=1|a2n|+n=1|a3n|+ (2)

has a finite sum M, then the additionPlanetmathPlanetmath in (1) can be performed in reverse , i.e.


Proof.  The assumptionPlanetmathPlanetmath on (2) implies that the sum of an arbitrary finite amount of the numbers |amn| is always M.  This means that (1) is absolutely convergent, and thus the order of summing is insignificant.

Note.  The series satisfying the assumptions of the theorem is often denoted by


and this may by interpreted to an arbitrary summing .  One can use e.g. the diagonal summing:

Title double series
Canonical name DoubleSeries
Date of creation 2013-03-22 16:32:54
Last modified on 2013-03-22 16:32:54
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Theorem
Classification msc 40A05
Classification msc 26A06
Synonym double series theorem
Related topic FourierSineAndCosineSeries
Related topic AbsoluteConvergenceOfDoubleSeries
Related topic PerfectPower
Defines diagonal summing