Question

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

Answer 1

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 = 1/3` gives us our expansion.

`(1+x)^{1/3} = 1 + \frac{x}3 - \frac{x^2}9 + \frac{5x^3}{81} + O(x^4)`