Translation of Index Variable of Summation/Infinite Series

Theorem

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

Let $\ds \sum_{\map R j} a_j$ denote a summation over $R$.

Let the fiber of truth of $R$ be infinite.

Then:

$\ds \sum_{\map R j} a_j = \sum_{\map R {c \mathop + j} } a_{c \mathop + j} = \sum_{\map R {c \mathop - j} } a_{c \mathop - j}$

where $c$ is an integer constant which is not dependent upon $j$.

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: Exercise $7$