Definition:Summation/Infinite

Definition
Let $\struct {S, +}$ 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 $\map R j$.

Then the precise meaning of $\ds \sum_{\map R j} a_j$ is:


 * $\ds \sum_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map R j \\ -n \mathop \le j \mathop < 0}} a_j} + \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$

provided that both limits exist.

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