Empty Sum - Significance of "terms" of An Empty Sum

Significance of "terms" of An Empty Sum

Since an empty sum by definition has no terms, it seems contradictory to talk about its terms; however in practice there almost always is an expression that describes the terms of a summation, even if the range of summation happens to be empty. Since this expression is never instantiated in an empty sum, its value is irrelevant; for instance the harmonic number

is perfectly well defined. However, the kind of values denoted by the summand is of importance for the value of the summation; for instance, an empty summation of elements of a vector space has as value the zero vector in that space, rather than the number 0. Even more important is the fact that the operation is summation; by contrast, the empty product—a product of no factors at all—has as value of one.