# Binomial approximation

The binomial approximation is useful for approximately calculating powers of sums of a small number and 1. It states that if ${\displaystyle x}$ is a real number close to 0 and ${\displaystyle \alpha }$ is a real number, then

${\displaystyle (1+x)^{\alpha }\approx 1+\alpha x.}$

This approximation can be obtained by using the binomial theorem and ignoring the terms beyond the first two.

By Bernoulli's inequality, the left-hand side of this relation is greater than or equal to the right-hand side whenever ${\displaystyle x>-1}$ and ${\displaystyle \alpha \geq 1}$.

## Derivation using linear approximation

The function

${\displaystyle f(x)=(1+x)^{\alpha }}$

is a smooth function for x near 0. Thus, standard linear approximation tools from calculus apply: one has

${\displaystyle f'(x)=\alpha (1+x)^{\alpha -1}}$

and so

${\displaystyle f'(0)=\alpha .}$

Thus

${\displaystyle f(x)\approx f(0)+f'(0)(x-0)=1+\alpha x.}$