Sum of Summations over Overlapping Domains/Infinite Series

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.


Let the fiber of truth of both $R$ and $S$ be infinite.


Then provided that any $3$ of the $4$ summations converge:

$\ds \sum_{\map R j} a_j + \sum_{\map S j} a_j = \sum_{\map R j \mathop \lor \map S J} a_j + \sum_{\map R j \mathop \land \map S j} a_j$

where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.


Proof




Sources