Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \left] 0 , + \infty \right[ ^ { \mathbb { N } }$ and $\ell \in \left] 0 , + \infty \right[$. Suppose that $\lim _ { n \rightarrow + \infty } \frac { u _ { n + 1 } } { u _ { n } } = \ell$. Prove that $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { u _ { n } } = \ell$. Prove that the result holds if $\ell = 0$ or $\ell = + \infty$. Deduce $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { n ! }$ and $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { \frac { n ^ { n } } { n ! } }$.
Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \left] 0 , + \infty \right[ ^ { \mathbb { N } }$ and $\ell \in \left] 0 , + \infty \right[$. Suppose that $\lim _ { n \rightarrow + \infty } \frac { u _ { n + 1 } } { u _ { n } } = \ell$. Prove that $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { u _ { n } } = \ell$. Prove that the result holds if $\ell = 0$ or $\ell = + \infty$. Deduce $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { n ! }$ and $\lim _ { n \rightarrow + \infty } \sqrt [ n ] { \frac { n ^ { n } } { n ! } }$.