Permutation of Indices of Summation/Infinite Series
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Theorem
Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers.
Let the fiber of truth of $R$ be infinite.
Let $\ds \sum_{\map R i} a_i$ be absolutely convergent.
Then:
- $\ds \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$
where:
- $\ds \sum_{\map R j} a_j$ denotes the summation over $a_j$ for all $j$ that satisfy the propositional function $\map R j$
- $\pi$ is a permutation on the fiber of truth of $R$.
Proof
This is a restatemtent of Manipulation of Absolutely Convergent Series: Permutation in the context of summations.
$\blacksquare$
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