# Definition:Summation/Summand/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 $A = \set {a_j: j \in \Z} \subseteq S$ be a set of elements of $S$.

Let $\map R j$ be a propositional function of $j$.

Let:

- $\ds \sum_{\map R j} a_j$

be an instance of a summation on $A$.

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**.

## Also known as

The **infinite summand** is also known as an **infinite set of summands**.

## Linguistic Note

The extensions **-and** and **-end** derive from the Latin gerundive forms which impart the meaning **that which must be ...** to a word.

Thus the word **summand**, and its synonym **addend**, literally mean: **that which must be summed (or added)**.

In natural language, the word **addendum** is more common than either, and similarly means **something which is to be added** (usually, by linguistic coincidence, to the **end**).

The archaic term **augend** has the same lingustic root as **augment**, which means **to make larger**. Hence **augend** is interpreted as **something which is to be made larger** by adding an **addend**.