Radius of convergence of a power series

Let \(F ( x ) = \sum _ { n = 0 } ^ { \infty } a _ { n } x ^ { n }\) be a power series with \(a _ { n } \in \mathbf { R }\) . Show that there exists a unique number \(\rho \geq 0\) such that \(F ( x )\) converges if \(| x | < \rho\) and \(F ( x )\) diverges if \(| x | > \rho\)