Answer
Diverges
Work Step by Step
\begin{aligned} \int_{2}^{\infty} x^{-\frac{1}{3}} d x &=\lim _{R \rightarrow \infty} \int_{2}^{R} x^{-\frac{1}{3}} d x \\
&=\lim _{R \rightarrow \infty}\left.\frac{x^{\frac{2}{3}}}{3}\right|_{2} ^{R} \\ &=\lim _{R \rightarrow \infty}\frac{3 R^{\frac{2}{3}}}{2}-\frac{3 \cdot 2^{\frac{2}{3}}}{2} \\
&=\infty\end{aligned}
Thus the integral diverges.
