Binomial approximation

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 \alpha is a real number, then

(1 %2B x)^\alpha \approx 1 %2B \alpha x.

This approximation can be obtained by using the binomial theorem 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 x>-1 and \alpha a non-negative integer, by Bernoulli's inequality.

Derivation using Mellin Transform

M(p) = \int^\infty_0 (1%2B\alpha x)^{-\gamma}x^{p-1}dx

Let y=\alpha x\,

M(p) = \alpha^{-p}\int^\infty_0 (1%2By)^{-\gamma}y^{p-1}dy

Let y=z/(1-z)

M(p) = \alpha^{-p}\int^1_0(1-z)^{\gamma-p-1}z^{p-1} dz

= \alpha^{-p}B(\gamma-p,p)\,

= \alpha^{-p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}.

Using the inverse Mellin transform:

(1%2B\alpha x)^{-\gamma}=\frac{1}{2\pi i}\int^{c%2Bi\infty}_{c-i\infty}(x\alpha)^{-p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}dp

Closing this integral to the left, which converges for |\alpha x|<1\,, we get:

(1%2B\alpha x)^{-\gamma}=\Sigma_{n=0}^{\infty}(\alpha x)^n \frac{(-1)^n}{n!}\frac{\Gamma(\gamma%2Bn)}{\Gamma(\gamma)}

=1-\alpha x \gamma%2B(1/2)(\alpha x)^2 (\gamma%2B1)\gamma-...\,

Derivation using Linear Approximation

f(x) = (1 %2B x)^{\alpha}.
f'(x) = \alpha (1 %2B x)^{\alpha - 1}.

When x = 0:

f'(0) = \alpha.

Using linear approximation:

f(x) \approx f(a) %2B f'(a)(x - a).
f(x) \approx f(0) %2B f'(0)(x - 0).
(1 %2B x)^{\alpha} \approx 1 %2B \alpha x.