Definition:Summation/Infinite

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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 an infinite number of values of $j$ satisfy the propositional function $R \left({j}\right)$.

Then the precise meaning of $\displaystyle \sum_{R \left({j}\right)} a_j$ is:

$\displaystyle \sum_{R \left({j}\right)} a_j = \left({\lim_{n \mathop \to \infty} \sum_{\substack {R \left({j}\right) \\ -n \mathop \le j \mathop < 0}} a_j}\right) + \left({\lim_{n \mathop \to \infty} \sum_{\substack {R \left({j}\right) \\ 0 \mathop \le j \mathop \le n} } a_j}\right)$

provided that both limits exist.

If either limit does fail to exist, then the infinite summation does not exist.


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).


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