real analysis - relations between the root test and the ratio test

relations between the root test and the ratio test

I know the theorem is correct if they are exist
$$
\lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}
\leq
\lim\inf\limits_{n\rightarrow \infty} (A_n)^{1/n}
\leq
\lim\sup\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \lim\sup\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}
$$

$$
$$

Here is the 1st question.

If
$$\lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $$ and
$$
\lim\sup\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}
$$

are $\infty$

then,
$$
\lim\inf\limits_{n\rightarrow \infty} (A_n)^{1/n}
$$
and
$$
\lim\sup\limits_{n\rightarrow \infty} (A_n)^{1/n}
$$

are $\infty$?

$$
$$

$$
$$

And 2nd question is
$$

\lim_{n\rightarrow \infty} \frac{|A_{n+1}|}{|A_n|} = \infty
$$
then $\lim_{n\rightarrow \infty} (A_n)^{1/n} = \infty$

$$
$$

$$
$$

Actually, 2nd question looks like easy, but I can't prove yet.

Could you please help me?

Thanks