The binomial approximation is useful for approximately calculating powers of numbers close to 1. It states that if x is a real number close to 0 and α is a real number, then
This approximation can be obtained by using the binomial theorem and ignoring the terms beyond the first two.
The lefthand side of this relation is always greater than or equal to the righthand side for x > − 1 and α a nonnegative integer, by Bernoulli's inequality.
Using the inverse Mellin transform:
Closing this integral to the left, which converges for , we get:
