Assuming "root 9 of n" is supposed to mean "the ninth root of n", that is [tex]\sqrt[9]n[/tex] we can use the ratio test, which says the series converges whenever
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{\sqrt[9]{n+1}}}{\frac{x^n}{\sqrt[9]n}}\right|<1[/tex]
We have
[tex]\displaystyle|x|\lim_{n\to\infty}\frac{\sqrt[9]n}{\sqrt[9]{n+1}}=|x|\sqrt[9]{\lim_{n\to\infty}\frac n{n+1}}=|x|<1[/tex]
which means the radius of convergence for this power series is 1.
