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Homedouble series

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# double series

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

$\displaystyle\sum_{{n=1}}^{\infty}|a_{{1n}}|+\sum_{{n=1}}^{\infty}|a_{{2n}}|+% \sum_{{n=1}}^{\infty}|a_{{3n}}|+\ldots$ | (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:

$a_{{11}}+a_{{12}}+a_{{21}}+a_{{13}}+a_{{22}}+a_{{31}}+\ldots$ |

Defines:

diagonal summing

Keywords:

absolutely convergent

Related:

FourierSineAndCosineSeries, AbsoluteConvergenceOfDoubleSeries, PerfectPower

Synonym:

double series theorem

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

40A05*no label found*26A06

*no label found*

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