Wednesday, 20 May 2015

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






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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