Exchange of Order of Summation/Finite and Infinite Series
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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 $R$ 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:
- $\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$
Proof
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