Exchange of Order of Summation

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Theorem

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ and $S: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be propositional functions on the set of integers.

Let $\displaystyle \sum_{R \left({i}\right)} x_i$ denote a summation over $R$.


Let the fiber of truth of both $R$ and $S$ be finite.


Then:

$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$


Finite and Infinite Series

Let the fiber of truth of $R$ be infinite.

Let the fiber of truth of $S$ be finite.


For all $j$ in the fiber of truth of $S$, let $\displaystyle \sum_{R \left({i}\right)} a_{i j}$ be convergent.


Then:

$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$


Infinite Series

Let the fiber of truth of both $R$ and $S$ be infinite.


Let:

$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} \left\vert{a_{i j} }\right\vert$

exist.


Then:

$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$


Proof


Example

Let the fiber of truth of both $R$ and $S$ be infinite.


Then it is not necessarily the case that:

$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$


Also known as

The word interchange can often be seen for exchange.


Sources