The binomial approximation is useful for approximately calculating powers

Exponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...

 of numbers close to 1. It states that if is a real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 close to 0 and is a real number, then

This approximation can be obtained by using the binomial theorem

Binomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

 and ignoring the terms beyond the first two.

The left-hand side of this relation is always greater than or equal to the right-hand side for and a non-negative integer, by Bernoulli's inequality

Bernoulli's inequality

In real analysis, Bernoulli's inequality is an inequality that approximates exponentiations of 1 + x.The inequality states that^r \geq 1 + rx\!...

.

Derivation using Mellin Transform

Let

Let y=z/(1-z)

Using the inverse Mellin transform

Mellin transform

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...

:



Closing this integral to the left, which converges for , we get:

Derivation using Linear Approximation

When x = 0:
Using linear approximation: