Does $\lim_{n \to \infty}a_n^{1/n} = 1$ imply $\lim_{n \to \infty} \frac{a_{n + 1}}{a_n} = 1$? Supose I have a sequence $\{a_n\}$ of positive real numbers such that $\lim\limits_{n \to \infty}a_n^{1/n} = 1$. Is it true that $\lim\limits_{n \to \infty} \frac{a_{n + 1}}{a_n} = 1$ or depends of the sequence that a choose?
 A: Let $r_n = \frac{a_{n + 1}}{a_n}$ and $s_n = a_n^{1/n}$. Then the rule is
$$ \liminf r_n \le \liminf s_n \le \limsup s_n \le \limsup r_n. $$
So if both limits exist then they are equal but $\lim s_n$ might exist where $\lim r_n$ might not.
A: Even if one exists, the other may not. Consider: $a_n=\frac{3+(-1)^n}{2^{n}}$
$$\lim_\limits{n\to\infty} \sqrt[n]{a_n}=\frac{1}{2}.$$
$$\lim_\limits{n\to\infty} \frac{a_{n+1}}{a_n}=\begin{cases} 1, n =\ odd \\ \frac{1}{4}, n= \ even \end{cases}.$$
A: Here a simple counterexample: let $a_n=1$ if $n$ is even, and $a_n=2$ if $n$ is odd. Clearly $\lim\limits_{n \mapsto \infty}a_n^{\frac{1}{n}} = 1,$ while $a_{n+1}/a_n$ oscillates between $2$ and $1/2,$ i.e., does not converge.
A: No, that is  false (see Reiner Martin's answer). On the other hand, the converse is true by Stolz-Cesaro Theorem: if $a_{n+1}/a_n\to L$ then
$$\lim_{n\to \infty}\ln(a_n^{1/n})=\lim_{n\to \infty}\frac{\ln(a_n)}{n}\stackrel{\text{SC}}{=}\lim_{n\to \infty}\frac{\ln(a_{n+1})-\ln(a_n)}{(n+1)-n}=\lim_{n\to \infty}\ln\left(\frac{a_{n+1}}{a_n}\right)=\ln(L)$$
that is $a_n^{1/n}\to L$.
