alternating sum

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


An alternating sum is also called an alternating series.

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 numberMathworldPlanetmathPlanetmath gives a negative number while raised to an even number gives a positive number (see: factors with minus sign), (-1)i essentially has the effect of turning the odd-indexed terms of the sequence negative but keeping their absolute valuesMathworldPlanetmathPlanetmathPlanetmath 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 π 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 infinityMathworldPlanetmathPlanetmath of operands. For example, the alternating factorialMathworldPlanetmath of n is computed by an alternating sum stopping at i=n.


  • 1 Tobias Dantzig, Number: The LanguagePlanetmathPlanetmath of Science, ed. Joseph Mazur. New York: Pi Press (2005): 166
Title alternating sum
Canonical name AlternatingSum
Date of creation 2013-03-22 17:35:30
Last modified on 2013-03-22 17:35:30
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Definition
Classification msc 11B25