# double series

If the double series

 $\displaystyle\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}=\sum_{n=1}^{\infty}a% _{1n}+\sum_{n=1}^{\infty}a_{2n}+\sum_{n=1}^{\infty}a_{3n}+\ldots$ (1)
 $\displaystyle\sum_{n=1}^{\infty}|a_{1n}|+\sum_{n=1}^{\infty}|a_{2n}|+\sum_{n=1% }^{\infty}|a_{3n}|+\ldots$ (2)

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

 $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}=\sum_{n=1}^{\infty}\sum_{m=1}^{% \infty}a_{mn}=\sum_{m=1}^{\infty}a_{m1}+\sum_{m=1}^{\infty}a_{m2}+\sum_{m=1}^{% \infty}a_{m3}+\ldots$

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

 $\sum_{m,n=1}^{\infty}a_{mn}$

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

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