Solution
= Solution
The generalized <binomial series> gives
$$
\boxed{(1+x)^{1/3}=\sum_{n=0}^\infty u_nx^n,\qquad
u_0=1,\quad u_n=\frac{(1/3)(1/3-1)\cdots(1/3-n+1)}{n!}\ (n\geq1)}.
$$
In particular,
$$
(1+x)^{1/3}=1+\frac{x}{3}-\frac{x^2}{9}+\frac{5x^3}{81}
-\frac{10x^4}{243}+\cdots.
$$
The recurrence $u_{n+1}=u_n(1/3-n)/(n+1)$ follows directly from the coefficient product. The <power series> has <radius of convergence> one and represents the real cube root for $|x|<1$.