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Alternating Sum
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