Binomial approximation

The binomial approximation is useful for approximately calculating powers of sums of a small number x and 1. It states that The binomial approximation is useful for approximately calculating powers of sums of a small number x and 1. It states that It is valid when | x | < 1 {displaystyle |x|<1} and | α x | ≪ 1 {displaystyle |alpha x|ll 1} where x {displaystyle x} and α {displaystyle alpha } may be real or complex numbers. The benefit of this approximation is that α {displaystyle alpha } is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics. The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever x > − 1 {displaystyle x>-1} and α ≥ 1 {displaystyle alpha geq 1} .