Empty sum

In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. By convention, the value of any empty sum of numbers is zero.

Let a₁, a₂, a₃,... be a sequence of numbers, and let

be the sum of the first m terms of the sequence. Then

for all m = 1,2,... provided that we use the following conventions: ${\displaystyle s_{1}=a_{1}}$ and ${\displaystyle s_{0}=0}$. In other words, a "sum" ${\displaystyle s_{1}}$ with only one term evaluates to that one term, while a "sum" ${\displaystyle s_{0}}$ with no terms evaluates to 0. Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas. Such "sums" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty sum is zero convention" is standard practice in mathematics and computer programming. For the same reason, the empty product is taken to be one, the neutral element for multiplication.

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