Assuming "root 9 of n" is supposed to mean "the ninth root of n", that is ![\sqrt[9]n](https://tex.z-dn.net/?f=%5Csqrt%5B9%5Dn)
we can use the ratio test, which says the series converges whenever
![\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{\sqrt[9]{n+1}}}{\frac{x^n}{\sqrt[9]n}}\right|](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%5Cfrac%7Bx%5E%7Bn%2B1%7D%7D%7B%5Csqrt%5B9%5D%7Bn%2B1%7D%7D%7D%7B%5Cfrac%7Bx%5En%7D%7B%5Csqrt%5B9%5Dn%7D%7D%5Cright%7C%3C1)
We have
![\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|](https://tex.z-dn.net/?f=%5Cdisplaystyle%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B%5Csqrt%5B9%5Dn%7D%7B%5Csqrt%5B9%5D%7Bn%2B1%7D%7D%3D%7Cx%7C%5Csqrt%5B9%5D%7B%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%20n%7Bn%2B1%7D%7D%3D%7Cx%7C%3C1)
which means the radius of convergence for this power series is 1.
We have
which means the radius of convergence for this power series is 1.