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:


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

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

Alternatively:


 * $$\sum \limits_{1 \le j \le n} a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$$

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


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

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

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 $$\sum \limits_{\Phi \left({j}\right)} a_j$$ is:


 * $$\sum \limits_{\Phi \left({j}\right)} a_j = \left({\lim_{n \to \infty} \sum_{\stackrel{R \left({j}\right)}{-n \le j < 0}} a_j}\right) + \left({\lim_{n \to \infty} \sum_{\stackrel{R \left({j}\right)}{0 \le j \le n}} a_j}\right)$$

provided that both limits exist. If either limit does fails 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.

Summand
The quantity after the summation sign is called the summand, or the set of summands.

Vacuous Summation
Take the sum:
 * $$\sum \limits_{\Phi \left({j}\right)} a_j$$

where $$\Phi \left({j}\right)$$ is a propositional function of $$j$$.

Suppose that there are no values of $$j$$ for which $$\Phi \left({j}\right)$$ is true.

Then $$\sum \limits_{\Phi \left({j}\right)} a_j$$ is defined as being $$0$$.

This summation is called a vacuous summation or vacuous sum.

This is most frequently seen in the form:
 * $$\sum_{j=m}^n a_j = 0$$

where $$m > n$$.

In this case, $$j$$ can not at the same time be both greater than or equal to $$m$$ and less than or equal to $$n$$.

Compare vacuous truth.

Historical Note
The notation $$\sum$$ was introduced by Joseph Fourier in 1820:
 * "Le signe $$\sum_{i=1}^{i=\infty}$$ 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 $$\sum_{i=1}^{i=\infty}$$ indicates that one must give to the whole number $$i$$ all the values $$1, 2, 3, \ldots$$, and take the sum of the terms.")

-- Refroidissement séculaire du globe terrestre, Bulletin des Sciences par la Société philomathique de Paris, series 3, 7 (1820), 58-70.