Definition:Summation/Inequality
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 $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.
The summation of $\tuple {a_1, a_2, \ldots, a_n}$ can be written:
- $\ds \sum_{1 \mathop \le j \mathop \le n} a_j = \paren {a_1 + a_2 + \cdots + a_n}$
Multiple Indices
Let $\ds \sum_{0 \mathop \le j \mathop \le n} a_j$ denote the summation of $\tuple {a_0, a_1, a_2, \ldots, a_n}$.
Summands with multiple indices can be denoted by propositional functions in several variables, for example:
- $\ds \sum_{0 \mathop \le i \mathop \le n} \paren {\sum_{0 \mathop \le j \mathop \le n} a_{i j} } = \sum_{0 \mathop \le i, j \mathop \le n} a_{i j}$
- $\ds \sum_{0 \mathop \le i \mathop \le n} \paren {\sum_{0 \mathop \le j \mathop \le i} a_{i j} } = \sum_{0 \mathop \le j \mathop \le i \mathop \le n} a_{i j}$
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).
Examples
From $1$ to $\pi$
Let $n = 3 \cdotp 1 4$.
Then:
- $\ds \sum_{1 \mathop \le j \mathop \le n} a_j = a_1 + a_2 + a_3$
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: $(1)$