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,
$$\mathrm{log}2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\mathrm{\dots}$$ |
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 number^{} 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 values^{} the same. Our previous example would thus be restated
$$\mathrm{log}2=\sum _{i=1}^{\mathrm{\infty}}{(-1)}^{i-1}\frac{1}{i}.$$ |
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 $\pi $ or fractions thereof, such as the Gregory series:
$$\frac{\pi}{4}=\sum _{i=0}^{\mathrm{\infty}}{(-1)}^{i}\frac{1}{2i+1}$$ |
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 $n$ is computed by an alternating sum stopping at $i=n$.
References
- 1 Tobias Dantzig, Number: The Language^{} 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 |