Answer
$1-\frac{2}{3}x-2\Sigma_{n=2}^{\infty}\frac{1.4.7......(3n-5)x^{n}}{3^{n}n!}$
and
$R=1$
Work Step by Step
${(1-x)^{2/3}}=(1+(-x))^{2/3}$
$=1-\frac{2}{3}x-2\Sigma_{n=2}^{\infty}\frac{1.4.7......(3n-5)x^{n}}{3^{n}n!}$
$$\lim\limits_{n \to \infty}|\dfrac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{\frac{1.4.7......(3n-5)(3n.2)x^{n+1}}{3^{n+1}(n+1)!}}{\frac{1.4.7......(3n-5)x^{n}}{3^{n}(n)!}}|$$
$=\lim\limits_{n \to\infty}|\frac{(3n-2)x}{(3n+3)}|$
$=|x|$
The series will converge when $|x|\lt 1$ so $R=1$.
