double series

Theorem.  If the double series

${\displaystyle \sum _{m=1}^{\mathrm{\infty }}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_{mn}={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_{1n}+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_{2n}+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_{3n}+\mathrm{\dots }$ (1)

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

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}|a_{1n}|+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}|a_{2n}|+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}|a_{3n}|+\mathrm{\dots }$ (2)

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

$$\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{a}_{mn}=\sum _{n=1}^{\mathrm{\infty }}\sum _{m=1}^{\mathrm{\infty }}{a}_{mn}=\sum _{m=1}^{\mathrm{\infty }}{a}_{m1}+\sum _{m=1}^{\mathrm{\infty }}{a}_{m2}+\sum _{m=1}^{\mathrm{\infty }}{a}_{m3}+\mathrm{\dots }$$

Proof.  The assumption on (2) implies that the sum of an arbitrary finite amount of the numbers $|a_{mn}|$ is always $\leqq M$.  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

$$\sum _{m,n=1}^{\mathrm{\infty }}{a}_{mn}$$

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

$$a_{11}+a_{12}+a_{21}+a_{13}+a_{22}+a_{31}+\mathrm{\dots }$$

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