 Solution

First suppose $L>0$. Taking logarithms,Divide by $n$. Since the summands tend to $\log L$, their Cesàro means also tend to $\log L$. Henceand exponentiation gives $a_n^{1/n}\to L$.

If $L=0$, then for every $\epsilon >0$ the ratios are eventually at most $\epsilon$, so $a_n\le C{\epsilon }^n$ for a suitable constant $C$. ThusLetting $\epsilon \downarrow 0$ proves the ratio limit implies root limit in all cases.