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 $S$.

Then the composite is called the sum of $\left({a_1, a_2, \ldots, a_n}\right)$, and is written:


 * $\displaystyle \sum_{j \mathop = 1}^n a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$

This can also be written:
 * $\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$

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.

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.

The sign $\sum$ is called the summation sign and sometimes referred to as sigma (as that's its name in Greek).

Also see

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

Historical Note
The notation $\sum$ was introduced by Joseph Fourier in 1820:
 * "Le signe $\displaystyle \sum_{i \mathop = 1}^{i \mathop = \infty}$ indique que l'on doit donner au nombre entier $i$ toutes les valeurs $1, 2, 3, \ldots$, et prendre la somme des termes."


 * ("The sign $\displaystyle \sum_{i \mathop = 1}^{i \mathop = \infty}$ indicates that one must give to the whole number $i$ all the values $1, 2, 3, \ldots$, and take the sum of the terms.")

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