Definition:Summation/Infinite

Definition

Let $\struct {S, +}$ 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 $\map R j$.

Then the precise meaning of $\ds \sum_{\map R j} a_j$ is:

$\ds \sum_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map R j \\ -n \mathop \le j \mathop < 0}} a_j} + \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$

provided that both limits exist.

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

Finite

Let the set of values of $j$ which satisfy the propositional function $\map R j$ be finite.

Then the summation $\ds \sum_{\map R j} a_j$ is described as being a finite summation.

Summand

Let an infinite number of values of $j$ satisfy $\map R j = \T$.

The set of elements $\set {a_j \in A: \map R j}$ is called an infinite 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 $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ 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: $(3)$