Definition:Summation/Inequality

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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 $S$.


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

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


Multiple Indices

Let $\displaystyle \sum_{0 \mathop \le j \mathop \le n} a_j$ denote the summation of $\left({a_0, a_1, a_2, \ldots, a_n}\right)$.


Summands with multiple indices can be denoted by propositional functions in several variables, for example:

$\displaystyle \sum_{0 \mathop \le i \mathop \le n} \left({\sum_{0 \mathop \le j \mathop \le n} a_{i j} }\right) = \sum_{0 \mathop \le i, j \mathop \le n} a_{i j}$


$\displaystyle \sum_{0 \mathop \le i \mathop \le n} \left({\sum_{0 \mathop \le j \mathop \le i} a_{i j} }\right) = \sum_{0 \mathop \le j \mathop \le i \mathop \le n} a_{i j}$


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).


Examples

From $1$ to $\pi$

Let $n = 3 \cdotp 1 4$.

Then:

$\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j = a_1 + a_2 + a_3$


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: pp. 58 – 70)


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


Sources