# absolute convergence of double series

Let us consider the double series

 $\displaystyle\sum_{i,j=1}^{\infty}u_{ij}$ (1)

of real or complex numbers $u_{ij}$.  Denote the row series $u_{k1}\!+\!u_{k2}\!+\ldots$ by $R_{k}$, the column series $u_{1k}\!+\!u_{2k}\!+\ldots$ by $C_{k}$ and the diagonal series $u_{11}\!+\!u_{12}\!+\!u_{21}\!+\!u_{13}\!+\!u_{22}\!+\!u_{31}\!+\ldots$ by DS.  Then one has the

Theorem.  All row series, all column series and the diagonal series converge absolutely and

 $\sum_{k=1}^{\infty}R_{k}\;=\;\sum_{k=1}^{\infty}C_{k}\;=\;\mbox{DS},$

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 $W_{1}\!+\!W_{2}\!+\ldots$ with

 $\sum_{j=1}^{\infty}|u_{kj}|=W_{k}$

is convergent.  An analogical condition may be formulated for the column series $C_{k}$.

Example.  Does the double series

 $\sum_{m=2}^{\infty}\sum_{n=3}^{\infty}n^{-m}$

converge?  If yes, determine its sum.

The column series $\displaystyle\sum_{m=2}^{\infty}\left(\frac{1}{n}\right)^{m}$ have positive terms and are absolutely converging geometric series having the sum

 $\frac{(1/n)^{2}}{1-1/n}\;=\;\frac{1}{n(n\!-\!1)}\;=\;\frac{1}{n\!-\!1}-\frac{1% }{n}\;=\;W_{n}.$

The series $W_{3}\!+\!W_{4}\!+\ldots$ is convergent, since its partial sum is a telescoping sum

 $\sum_{n=3}^{N}W_{n}\;=\;\sum_{n=3}^{N}\left(\frac{1}{n\!-\!1}-\frac{1}{n}% \right)\;=\;\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}% \right)+\left(\frac{1}{4}-\frac{1}{5}\right)+\ldots+\left(\frac{1}{N\!-\!1}-% \frac{1}{N}\right)$

equalling simply $\frac{1}{2}\!-\!\frac{1}{N}$ and having the limit $\frac{1}{2}$ as  $N\to\infty$.  Consequently, the given double series converges and its sum is $\frac{1}{2}$.

Title absolute convergence of double series AbsoluteConvergenceOfDoubleSeries 2013-03-22 18:46:45 2013-03-22 18:46:45 pahio (2872) pahio (2872) 7 pahio (2872) Definition msc 40A05 DoubleSeries DiagonalSumming row series column series diagonal series