# Exchange of Order of Summation/Example

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## 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 it is not necessarily the case that:

- $\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$

## Proof

This theorem requires a proof.In particular: No idea at the moment. Have to find a series of $a_{i j}$ which is conditionally convergent for a start.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 $8$