Exchange of Order of Summation with Dependency on Both Indices/Infinite Series

Theorem

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

Let $S: \Z \times \Z \to \set {\T, \F}$ be a propositional functions on the Cartesian product of the set of integers with itself.

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

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

Let:

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

exist.

Then:

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

where:

$\map {S'} j$ denotes the propositional function:

there exists an $i$ such that both $\map R i$ and $\map S {i, j}$ hold

$\map {R'} {i, j}$ denotes the propositional function:

both $\map R i$ and $\map S {i, j}$ hold.

Proof

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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