Binomial approximation

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

(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 $x>-1$ and $\alpha \geq 1$ .

Derivation using linear approximation

The function

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

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

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

and so

f'(0)=\alpha .

Thus

f(x)\approx f(0)+f'(0)(x-0)=1+\alpha x.