Definition:Summation/Propositional Function

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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 $S$.


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.


Such an operation on an ordered tuple is known as a summation.


Note that the definition by inequality form $1 \le j \le n$ is a special case of such a propositional function.

Also note that the definition by index form $\displaystyle \sum_{j \mathop = 1}^n$ is merely another way of writing $\displaystyle \sum_{1 \mathop \le j \mathop \le n}$.

Hence all instances of a summation can be expressed in terms of a propositional function.


Iverson's Convention

Let $\displaystyle \sum_{R \left({j}\right)} a_j$ be the summation over all $a_j$ such that $j$ satisfies $R$.


This can also be expressed:

$\displaystyle \sum_{j \mathop \in \Z} a_j \left[{R \left({j}\right)}\right]$

where $\left[{R \left({j}\right)}\right]$ is Iverson's convention.


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