Let $r$ the radius of convergence of the power series $\sum a_n(x-x_0)^n.$ Prove that if $r\in \mathbb{R^+}$, then $r=1/L$, where $L$ is the greatest subsequential limit of the bounded sequence $(\sqrt[n]{|a_n|})$. Hence, $r=1/(\lim \sup \sqrt[n]{|a_n|}$).
Any help on how to solve this? In class it was presented to us that $r=1/L$, where $L = \lim \sqrt[p]{|a_n|}$. This result supposes that the sequence $(\sqrt[p]{|a_n|}) $ converges, and if so it will be in fact equal to $\lim \sup$. But how to proceed with more generality, without assuming the convergence of the series in order to prove that $r=1/(\lim \sup \sqrt[n]{|a_n|}$)?
Thanks in advance