# Binomial Theorem/Approximations/2nd Order

## Theorem

Consider the General Binomial Theorem:

$\paren {1 + x}^\alpha = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots$

When $x$ is sufficiently small that $x^3$ can be neglected then:

$\paren {1 + x}^\alpha \approx 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } 2 x^2$

and the error is of the order of $\dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 3} } 6 x^3$