Definition:Summation/Propositional Function/Iverson's Convention

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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.


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.