Definition:Summation/Indexed

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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 $S$.


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

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


Summand

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


Notation

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


Also see

  • Results about summations can be found here.


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