relations between the root test and the ratio test
I know the theorem is correct if they are exist
lim
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
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