Exchange of Order of Summation/Finite and Infinite Series

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 $R$ 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:

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

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