How to show that $\lim_{n \to \infty} a_n^{1/n} = l$?
Suppose that $a_n > 0$, $n \in \mathbb{N}$. Suppose that $$\lim_{n \to \infty} a_{n+1}/a_n =l$$.
How to show that
$$\lim_{n \to \infty} a_n^{1/n} = l \;?$$
My solution: let $$b_n = a_{n+1}/a_n$$
Then
$$b_1 b_2 \cdots b_{n-1} = a_{n}/a_1$$.
Do we have $\lim_{n \to \infty} (b_1 \cdots b_{n-1})^{1/n} = l\;?$
$b_n \to l \to \ln b_n \to \ln l \to \dfrac{\ln b_1 + \ln b_2 +\cdots \ln b_n}{n} \to \ln l$ by Cesaro theorem, and you are done.