Change of Index Variable of Summation

Theorem

$\ds \sum_{\map R i} a_i = \sum_{\map R j} a_j$

where $\ds \sum_{\map R i} a_i$ denotes the summation over $a_i$ for all $i$ that satisfy the propositional function $\map R i$.

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: $(5)$