Definition:Summation/Indexed

Standard Number System
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $a, b \in \Z$ be integers.

Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$.

Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping.

The indexed summation of $f$ from $a$ to $b$ is recursively defined and denoted:


 * $\displaystyle \sum_{k \mathop = a}^b f(k) = \begin{cases} 0 & : b < a \\ \left( \displaystyle \sum_{k \mathop = a}^{b-1} f(k) \right) + f(b) & : b \geq a\end{cases}$

Abelian Group
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 $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

The composite is called the indexed summation of $\left({a_1, a_2, \ldots, a_n}\right)$, and is written:


 * $\displaystyle \sum_{j \mathop = 1}^n a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$

Also see

 * Definition:Summation over Finite Set

Also see

 * Indexed Summation does not Change under Permutation