# Binomial approximations

An expression of the form:

$(1 + x)^{n}$

where $$|x| < 1$$ and $$n$$ is a non-integer, can be approximated using sufficiently many terms of the expression:

$1 + nx + n(n-1)\frac{x^{2}}{2!} + n(n-1)(n-2)\frac{x^{3}}{3!} + ...$

### Examples

Expression $$1$$ $$x$$ $$x^{2}$$ $$x^{3}$$
$$(1 + x)^{\frac{1}{2}}$$ $$1$$ $$\frac{1}{2}$$ $$-\frac{1}{8}$$ $$\frac{1}{16}$$
$$1 - \sqrt{1 - x}$$ $$0$$ $$\frac{1}{2}$$ $$-\frac{1}{8}$$ $$\frac{1}{16}$$
$$(1+x)^{-\frac{1}{2}}$$ $$1$$ $$-\frac{1}{2}$$ $$\frac{3}{8}$$ $$-\frac{5}{16}$$
$$(1 - 2x^{2})^{-\frac{1}{2}}$$ $$1$$ $$0$$ $$1$$ $$0$$