1. ## Coefficient of x^n

What is the coefficient of $x^n$ in the power series form of $\sqrt[3]{1-2x}$?

My attempt:

$\sqrt[3]{1-2x}=(1-2x)^\frac{1}{3}=\sum_{n\geq 0}\dbinom{\frac{1}{3}}{n}(-2x)^n$

$\dbinom{\frac{1}{3}}{n}=\frac{\frac{1}{3}.\frac{-2}{3}.\frac{-5}{3}...\frac{-3n+4}{3}}{n!}$

How do I simplify this and find the coefficient of $x^n$?

2. ## Re: Coefficient of x^n

What is the coefficient of $x^n$ in the power series form of $\sqrt[3]{1-2x}$?
$\textrm{coef}\;(x^n)=(-2)^n\binom{1/3}{n}\quad (\forall n\geq 0)$ . For $n\geq 2$ we can simplify to obtain
$\textrm{coef}\;(x^n)=\ldots=-\frac{2^n}{3^n}\cdot \dfrac{2\cdot 5\cdot \ldots\cdot (3n-4)}{n!}$