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double series
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(Theorem)
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Theorem. If the double series
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(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
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(2) |
has a finite sum $M$ , then the addition in (1) can be performed in reverse order, i.e. $$\sum_{m=1}^\infty\sum_{n=1}^\infty a_{mn} = \sum_{n=1}^\infty\sum_{m=1}^\infty a_{mn} = \sum_{m=1}^\infty a_{m1}+\sum_{m=1}^\infty a_{m2}+\sum_{m=1}^\infty a_{m3} +\ldots$$
Proof. The assumption on (2) implies that the sum of an arbitrary finite amount of the numbers $|a_{mn}|$ 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 $$\sum_{m,n=1}^\infty a_{mn}$$ and this may by interpreted to mean an arbitrary summing order. One can use e.g. the diagonal summing: $$a_{11}+a_{12}+a_{21}+a_{13}+a_{22}+a_{31}+\ldots$$
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"double series" is owned by PrimeFan. [ owner history (3) ]
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Cross-references: order, absolutely convergent, numbers, implies, proof, addition, sum, finite, absolute values, series, convergent, converges, theorem
There are 4 references to this entry.
This is version 3 of double series, born on 2007-01-09, modified 2009-01-26.
Object id is 8731, canonical name is DoubleSeries.
Accessed 3781 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) |
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Pending Errata and Addenda
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