double seriesDoubleSeries2007-01-09 11:33:112009-01-26 15:43:04Theoremconvergent seriesdiagonal summingabsolutely convergent% this is the default PlanetMath preamble. as your knowledge
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\textbf{Theorem.}\, If the {\em double series}
\begin{align}
\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
\end{align}
converges and if it remains convergent when the \PMlinkescapetext{terms} of the partial series are replaced with their absolute values, i.e. if the series
\begin{align}
\sum_{n=1}^\infty|a_{1n}|+\sum_{n=1}^\infty|a_{2n}|+\sum_{n=1}^\infty|a_{3n}|
+\ldots
\end{align}
has a finite sum $M$, then the addition in (1) can be performed in reverse \PMlinkescapetext{order}, i.e.
$$\sum_{m=1}^\infty\sum_{n=1}^\infty a_{mn} = \sum_{n=1}^\infty\sum_{m=1}^\infty a_{mn} =
\sum_{m=1}^\infty a_{m1}+\sum_{m=1}^\infty a_{m2}+\sum_{m=1}^\infty a_{m3}
+\ldots$$
{\em 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.
\textbf{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 \PMlinkescapetext{mean} an arbitrary summing \PMlinkescapetext{order}.\, One can use e.g. the {\em diagonal summing}:
$$a_{11}+a_{12}+a_{21}+a_{13}+a_{22}+a_{31}+\ldots$$