Permutation of Indices of Summation/Infinite Series

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