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,

$$\log 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\mathrm{\dots }$$

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

$$\log 2=\sum _{i=1}^{\mathrm{\infty }}{(-1)}^{i-1}\frac{1}{i}.$$

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 $i=n$.