It's known that root test is stronger than ratio test, it means whenever ratio test works, then so does root test, but the converse is not true, I've been looking for a proof and I found this paper:
I really cannot understand what's happened in the two last lines, how did the author conclude that:
$$\left(L-\epsilon\right)^{n}\cdot\frac{a_{N}}{\left(L-\epsilon\right)^{N}}<a_{n}<\left(L+\epsilon\right)^{n}\cdot\frac{a_{N}}{\left(L+\epsilon\right)^{N}}$$
and from this it's concluded :
$$\left(L-\epsilon\right)<\sqrt[n]{a_{n}}<\left(L+\epsilon\right)$$
The last line doesn't seem correct. What we have is that $$(L-\epsilon)\sqrt[n]{\frac{a_N}{(L-\epsilon)^N}}<\sqrt[n]{a_n}<(L+\epsilon)\sqrt[n]{\frac{a_N}{(L+\epsilon)^N}}$$ So, taking $\limsup$ and $\liminf$, we get $$L-\epsilon\leq \liminf\sqrt[n]{a_n}\leq \limsup \sqrt[n]{a_n} \leq L+\epsilon.$$ Since this is true for all $\epsilon>0$, we conclude that $\liminf\sqrt[n]{a_n}= \limsup \sqrt[n]{a_n}=L$. Hence $\lim\sqrt[n]{a_n}=L$.