Definition:Summation

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


Definition by Index

The composite is called the summation of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:

$\displaystyle \sum_{j \mathop = 1}^n a_j = \paren {a_1 + a_2 + \cdots + a_n}$


Definition by Inequality

The summation of $\left({a_1, a_2, \ldots, a_n}\right)$ can be written:

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


Definition by Propositional Function

Let $R \left({j}\right)$ be a propositional function of $j$.

Then we can write the summation as:

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


If more than one propositional function is written under the summation sign, they must all hold.


Infinite

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

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

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

provided that both limits exist.

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


Index Variable

Consider the summation, in either of the three forms:

$\displaystyle \sum_{j \mathop = 1}^n a_j \qquad \sum_{1 \mathop \le j \mathop \le n} a_j \qquad \sum_{R \left({j}\right)} a_j$


The variable $j$, an example of a bound variable, is known as the index variable of the summation.


Summand

The set of elements $\set {a_j \in S: 1 \le j \le n, \map R j}$ is called the summand.


Vacuous Summation

Take the summation:

$\displaystyle \sum _{\Phi \left({j}\right)} a_j$

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 $\displaystyle \sum_{\Phi \left({j}\right)} a_j$ is defined as being $0$.

This summation is called a vacuous summation.


This is because:

$\forall a: a + 0 = a$

where $a$ is a number.

Hence for all $j$ for which $\Phi \left({j}\right)$ is false, the sum is unaffected.


This is most frequently seen in the form:

$\displaystyle \sum_{j \mathop = m}^n a_j = 0$

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 $n$.


Some sources consider such a treatment as abuse of notation.


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 $\mathsf{Pr} \infty \mathsf{fWiki}$ for immediate clarity.


Notation

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


Also see


Historical Note

The notation $\sum$ for a summation was famously 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.)
-- 1820: Refroidissement séculaire du globe terrestre (Bulletin des Sciences par la Société Philomathique de Paris Vol. 3, 7: 58 – 70)


However, some sources suggest that it was in fact first introduced by Euler.


Sources