absolute convergence of double series
Let us consider the double series
(1) |
of real or complex numbers . Denote the row series by , the column series by and the diagonal series
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:
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The diagonal series converges absolutely.
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There exists a positive number such that every finite sum of the numbers is .
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The row series converge absolutely and the series with
is convergent. An analogical condition may be formulated for the column series .
The column series have positive terms and are absolutely converging geometric series having the sum
The series is convergent, since its partial sum is a telescoping sum
equalling simply and having the limit as . Consequently, the given double series converges and its sum is .
Title | absolute convergence of double series |
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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 |