# How do you use the binomial series to expand f(x)=(1+x)^(1/3 )?

Jul 4, 2016

You cannot apply the usual binomial expansion (which is not applicable for non-integral rationals) here. Instead, use the binomial theorem for any index, stated as follows:

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

Just plugging in $n = \frac{1}{3}$ gives us our expansion.

${\left(1 + x\right)}^{\frac{1}{3}} = 1 + \setminus \frac{x}{3} - \setminus \frac{{x}^{2}}{9} + \setminus \frac{5 {x}^{3}}{81} + O \left({x}^{4}\right)$