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
 A: Both tests are for series with positive terms; or you should put absolute values around $A_n$ and $A_{n+1}$ everywhere. Otherwise you'll have a problem taking the roots. And division by $0$ is generally frowned upon. 
Assuming all the terms are positive, it is true that 
$$\liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} 
\leq 
\liminf\limits_{n\rightarrow \infty} (A_n)^{1/n} 
\leq 
\limsup\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \limsup_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}$$
in the extended sense: the limits are allowed to take value $+\infty$. (By the way, this is common with $\liminf $ and $\limsup$ anyway. 
The proof is not really different from the finite case. Here's a proof of the first inequality. Let $b$ be a number such that $b< \liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $. Then we have $A_{n+1}>bA_n$ for all sufficiently large $n$. Applying this iteratively, you'll get a lower bound of the form $A_n\ge C b^n$ for all sufficiently large $n$. This implies $\liminf\limits_{n\rightarrow \infty} (A_n)^{1/n} \ge b$. Since $b$ was an arbitrary number such that $b< \liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} $, the conclusion $\liminf\limits_{n\rightarrow \infty} (A_n)^{1/n}\ge \liminf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}$ follows. 
Note that it makes no difference whether these limits are finite or not; the argument works the same.
