# Exchange of Order of Summation

## 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:

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

## 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.