Exchange of Order of Summation

Theorem

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

Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.

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

Then:

$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$

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 $\ds \sum_{\map R i} a_{i j}$ be convergent.

Then:

$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$

Infinite Series

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

Let:

$\ds \sum_{\map R i} \sum_{\map S j} \size {a_{i j} }$

exist.

Then:

$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$

Proof

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Example

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

Then it is not necessarily the case that:

$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$

Also known as

The word interchange can often be seen for exchange.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: $(7)$