absolute convergence of double series
Let us consider the double series^{}
$\sum _{i,j=1}^{\mathrm{\infty}}}{u}_{ij$  (1) 
of real or complex numbers^{} ${u}_{ij}$. Denote the row series ${u}_{k1}+{u}_{k2}+\mathrm{\dots}$ by ${R}_{k}$, the column series ${u}_{1k}+{u}_{2k}+\mathrm{\dots}$ by ${C}_{k}$ and the diagonal series ${u}_{11}+{u}_{12}+{u}_{21}+{u}_{13}+{u}_{22}+{u}_{31}+\mathrm{\dots}$
by DS. Then one has the
Theorem. All row series, all column series and the diagonal series converge absolutely and
$$\sum _{k=1}^{\mathrm{\infty}}{R}_{k}=\sum _{k=1}^{\mathrm{\infty}}{C}_{k}=\text{DS},$$ 
if one of the following conditions is true:

•
The diagonal series converges absolutely.

•
There exists a positive number $M$ such that every finite sum of the numbers ${u}_{ij}$ is $\leqq M$.

•
The row series ${R}_{k}$ converge absolutely and the series ${W}_{1}+{W}_{2}+\mathrm{\dots}$ with
$$\sum _{j=1}^{\mathrm{\infty}}{u}_{kj}={W}_{k}$$ is convergent. An analogical condition may be formulated for the column series ${C}_{k}$.
Example. Does the double series
$$\sum _{m=2}^{\mathrm{\infty}}\sum _{n=3}^{\mathrm{\infty}}{n}^{m}$$ 
converge? If yes, determine its sum.
The column series $\sum _{m=2}^{\mathrm{\infty}}}{\left({\displaystyle \frac{1}{n}}\right)}^{m$ have positive terms and are absolutely converging geometric series^{} having the sum
$$\frac{{(1/n)}^{2}}{11/n}=\frac{1}{n(n1)}=\frac{1}{n1}\frac{1}{n}={W}_{n}.$$ 
The series ${W}_{3}+{W}_{4}+\mathrm{\dots}$ is convergent, since its partial sum is a telescoping sum
$$\sum _{n=3}^{N}{W}_{n}=\sum _{n=3}^{N}\left(\frac{1}{n1}\frac{1}{n}\right)=\left(\frac{1}{2}\frac{1}{3}\right)+\left(\frac{1}{3}\frac{1}{4}\right)+\left(\frac{1}{4}\frac{1}{5}\right)+\mathrm{\dots}+\left(\frac{1}{N1}\frac{1}{N}\right)$$ 
equalling simply $\frac{1}{2}\frac{1}{N}$ and having the limit $\frac{1}{2}$ as $N\to \mathrm{\infty}$. Consequently, the given double series converges and its sum is $\frac{1}{2}$.
Title  absolute convergence of double series 

Canonical name  AbsoluteConvergenceOfDoubleSeries 
Date of creation  20130322 18:46:45 
Last modified on  20130322 18:46:45 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  7 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 40A05 
Related topic  DoubleSeries 
Related topic  DiagonalSumming 
Defines  row series 
Defines  column series 
Defines  diagonal series 