Empty Sum - Summation Convention

Summation Convention

Empty sums, or even sums of a single term, do not play a role in the definition of addition, since that operation requires exactly two operands. The need to consider empty sums arises with summation: the process of "adding together" a collection of values that can have an arbitrary size. For a finite collection of two or more numbers, the commutative and associative laws of addition imply that every expression formed using addition only, and in which all members of the collection appear exactly once as operand, has the same value; this defines the sum of the collection. For infinite collections of values this definition does not apply, as no (finite) expression can combine them all using addition operations; the notion of a series can be used to attach a definite sum to some infinite collections, but this requires more than addition only, notably some notion of limit.

This leaves the cases of collections with less than two elements. One could decide to leave the sum of such collections undefined, on the grounds that there are too few values to perform any addition. For various reasons it is however useful to not make such an exception, and define the sum of any finite collection of values. Doing so should be done without invalidating the usual properties of summation, notably the fact that adjoining a new value x to a collection adds x to the sum of the collection. This property then implies that the sum of a collection containing a single value v is v, and that the sum of a collection of no values at all is 0, the neutral element for addition. An alternative approach is to define the sum of a finite sequence of values by induction on its length, with as starting case the empty sequence whose sum is 0. Both approaches define the same notion of sum, and the latter does so without making any separate definition for an empty sum.