Definition:Summation/Propositional Function/Iverson's Convention

Definition

Let $\ds \sum_{\map R j} a_j$ be the summation over all $a_j$ such that $j$ satisfies $R$.

This can also be expressed:

$\ds \sum_{j \mathop \in \Z} a_j \sqbrk {\map R j}$

where $\sqbrk {\map R j}$ 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 $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ 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 $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ 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

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: $(17)$