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单词 DoubleSeries
释义

double series


Theorem.  If the double series

m=1n=1amn=n=1a1n+n=1a2n+n=1a3n+(1)

converges and if it remains convergentMathworldPlanetmathPlanetmath when the of the partial series are replaced with their absolute valuesMathworldPlanetmathPlanetmathPlanetmath, i.e. if the series

n=1|a1n|+n=1|a2n|+n=1|a3n|+(2)

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

m=1n=1amn=n=1m=1amn=m=1am1+m=1am2+m=1am3+

Proof.  The assumptionPlanetmathPlanetmath on (2) implies that the sum of an arbitrary finite amount of the numbers |amn| is always 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

m,n=1amn

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

a11+a12+a21+a13+a22+a31+
Titledouble series
Canonical nameDoubleSeries
Date of creation2013-03-22 16:32:54
Last modified on2013-03-22 16:32:54
OwnerPrimeFan (13766)
Last modified byPrimeFan (13766)
Numerical id6
AuthorPrimeFan (13766)
Entry typeTheorem
Classificationmsc 40A05
Classificationmsc 26A06
Synonymdouble series theorem
Related topicFourierSineAndCosineSeries
Related topicAbsoluteConvergenceOfDoubleSeries
Related topicPerfectPower
Definesdiagonal summing
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更新时间:2025/5/4 20:43:37