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

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