Definition:Summation

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 $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

Definition by Propositional Function
If $\Phi \left({j}\right)$ is a propositional function of $j$, then we can write:


 * $\displaystyle \sum_{\Phi \left({j}\right)} a_j = \text{ The sum of all } a_j \text{ such that } \Phi \left({j}\right) \text{ holds}$.

Such an operation on an ordered tuple is known as a summation.

Note that $1 \le j \le n$ is in fact a special case of such a propositional function, and that $\displaystyle \sum_{j\mathop = 1}^n$ is merely another way of writing $\displaystyle \sum_{1 \mathop \le j \mathop \le n}$.

Thus, when it comes down to it, all instances of a summation can be expressed in terms of a propositional function.

Infinite
If 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:


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

provided that both limits exist.

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

Note also that if more than one propositional function is written under the summation sign, they must all hold.

Also known as
It is common for the term sum to be used for summation.

However, despite the fact that summation is longer, and a less-aesthetic neologism for sum, the term summation is preferred on for immediate clarity.

Also see

 * Definition:Product Notation (Algebra)
 * Definition:Composite (Abstract Algebra)
 * Definition:Series