PlanetMath: double series

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

double series

(Theorem)

Theorem. If the double series

$\displaystyle \sum_{m=1}^\infty\sum_{n=1}^\infty a_{mn} = \sum_{n=1}^\infty a_{1n}+\sum_{n=1}^\infty a_{2n}+\sum_{n=1}^\infty a_{3n} +\ldots$

(1)

converges and if it remains convergent when the terms of the partial series are replaced with their absolute values, i.e. if the series

$\displaystyle \sum_{n=1}^\infty\vert a_{1n}\vert+\sum_{n=1}^\infty\vert a_{2n}\vert+\sum_{n=1}^\infty\vert a_{3n}\vert +\ldots$

(2)

has a finite sum

, then the addition in (1) can be performed in reverse order, i.e.

$\displaystyle \sum_{m=1}^\infty\sum_{n=1}^\infty a_{mn} = \sum_{n=1}^\infty\sum... ...m_{m=1}^\infty a_{m1}+\sum_{m=1}^\infty a_{m2}+\sum_{m=1}^\infty a_{m3} +\ldots$

Proof. The assumption on (2) implies that the sum of an arbitrary finite amount of the numbers $\vert a_{mn}\vert$ is always $\leqq M$ . 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

$\displaystyle \sum_{m,n=1}^\infty a_{mn}$

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

$\displaystyle a_{11}+(a_{12}+a_{21})+(a_{13}+a_{22}+a_{31})+\ldots$

"double series" is owned by PrimeFan. [ owner history (3) ]

(view preamble)

See Also: Fourier sine and cosine series

Also defines:

diagonal summing

Keywords:

absolutely convergent

This object's parent.

Attachments:: Weierstrass double series theorem (Theorem) by pahio

Log in to rate this entry.
(view current ratings)

Cross-references: mean, order, absolutely convergent, numbers, implies, addition, sum, finite, absolute values, series, convergent, converges
There are 3 references to this entry.

This is version 1 of double series, born on 2007-01-09.
Object id is 8731, canonical name is DoubleSeries.
Accessed 1773 times total.

Classification:

AMS MSC:	40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
	26A06 (Real functions :: Functions of one variable :: One-variable calculus)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact