1
$\begingroup$

Possible Duplicate:
Inequality involving $\limsup$ and $\liminf$
limit of $\frac{a_{n+1}}{a_n}$

Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$ and similarly $\liminf|s_n|^{1\over n}\ge \liminf|{s_{n+1}\over s_n}|$. I have no idea where to start. I tried to show the inequality through subsequence but still don't quite get where to start. Any explaination how to link to the concept of related topic would be appreciated.

  • 0
    @MartinSleziak: Much appreciated!2012-07-23

1 Answers 1

1

Here’s a start for you.

For convenience we may suppose that the $s_n$ are non-negative and omit the absolute value signs. Suppose that $\limsup_ns_n^{1\over n}>\limsup_n\,{s_{n+1}\over s_n}\;.$ Let $b=\frac12\left(\limsup_ns_n^{1\over n}+\limsup_n\,{s_{n+1}\over s_n}\right)\;;$ then there is an $n_0\in\Bbb N$ such that $\frac{s_{n+1}}{s_n} for all $n\ge n_0$. Thus, for all $n>n_0$ we have $s_n. Let $a=s_{n_0}b^{-n_0}$, so that $s_n for $n>n_0$. Then $s_n^{1/n} for $n>n_0$.

  1. What is $\lim\limits_{n\to\infty}a^{1/n}$?

  2. How does $b$ compare with $\limsup\limits_n\,s_n^{1/n}$?