Answer
$\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{6n+5}}{2n+1}$, $R=1$
Work Step by Step
$f(x)=x^{2}tan^{-1}(x^{3})=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{6n+5}}{2n+1}$
$\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{\frac{x^{6(n+1)+5}}{2(n+1)+1}}{\frac{x^{6n+5}}{2n+1}}|$
$=|x^{6}|\lt 1$
The given series converges with $R=1$