Definition:Summation/Infinite

Definition
Let $\left({S, +}\right)$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers.

Let an infinite number of values of $j$ satisfy the propositional function $\Phi \left({j}\right)$.

Then the precise meaning of $\displaystyle \sum_{\Phi \left({j}\right)} a_j$ is:


 * $\displaystyle \sum_{\Phi \left({j}\right)} a_j = \left({\lim_{n \mathop \to \infty} \sum_{\substack {\Phi \left({j}\right) \\ -n \mathop \le j \mathop < 0}} a_j}\right) + \left({\lim_{n \mathop \to \infty} \sum_{\substack {\Phi \left({j}\right) \\ 0 \mathop \le j \mathop \le n} } a_j}\right)$

provided that both limits exist.

If either limit does fail to exist, then the infinite summation does not exist.