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.
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