An alternating sum is a sequence of arithmetic operations in which each addition is followed by a subtraction, and viceversa, applied to a sequence of numerical entities. For example,

Alternating sums are often expressed in summation notation with the iterated expression involving multiplication by negative one raised to the iterator. Since a negative number raised to an odd number gives a negative number while raised to an even number gives a positive number (see: factors with minus sign), essentially has the effect of turning the odd-indexed terms of the sequence negative but keeping their absolute values the same. Our previous example would thus be restated

If the operands in an alternating sum decrease in value as the iterator increases, and approach zero, then the alternating sum converges to a specific value. This fact is used in many of the best-known expression for or fractions thereof, such as the Gregory series:

Other constants also find expression as alternating sums, such as Cahen's constant.

An alternating sum need not necessarily involve an infinity of operands. For example, the alternating factorial of is computed by an alternating sum stopping at .

Bibliography

1

Tobias Dantzig, Number: The Language of Science, ed. Joseph Mazur. New York: Pi Press (2005): 166