The ratio limit bounds the root limit

Let \(a _ { n } \neq 0\) and assume that \(\operatorname* { l i m } _ { n \to \infty } { \frac { | a _ { n + 1 } | } { | a _ { n } | } } = L\) . Show that \(\operatorname* { l i m } _ { n \to \infty } \sqrt [ n ] { | a _ { n } | } = L\) . In particular, this shows that when applicable, the ratio test can be used to calculate the radius of convergence of a power series.