absolute convergence of double series

Let us consider the double seriesMathworldPlanetmath

i,j=1uij (1)

of real or complex numbersPlanetmathPlanetmath uij.  Denote the row series uk1+uk2+ by Rk, the column series u1k+u2k+ by Ck and the diagonal series u11+u12+u21+u13+u22+u31+ by DS.  Then one has the

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


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 |uij| is M.

  • The row series Rk converge absolutely and the series W1+W2+ with


    is convergent.  An analogical condition may be formulated for the column series Ck.

Example.  Does the double series


converge?  If yes, determine its sum.

The column series m=2(1n)m have positive terms and are absolutely converging geometric seriesMathworldPlanetmath having the sum


The series W3+W4+ is convergent, since its partial sum is a telescoping sum


equalling simply 12-1N and having the limit 12 as  N.  Consequently, the given double series converges and its sum is 12.

Title absolute convergence of double series
Canonical name AbsoluteConvergenceOfDoubleSeries
Date of creation 2013-03-22 18:46:45
Last modified on 2013-03-22 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