Ratio test for sequences, the other direction

Consider the sequence $$\frac{1}{1},-\frac{1}{1},\frac{1}{2},-\frac{1}{2},\frac{1}{3},-\frac{1}{3},\ldots$$

The terms go to zero, but consecutive terms have ratio that is $1$ in absolute value, half of the time.

Put $x_n = 1/n$ if $n$ is even and $x_n = 2/n$ if $n$ is odd. Then $$\left \lvert \frac{x_{n+1}}{x_n} \right \rvert = 2^{(-1)^n}\frac{n}{n+1}$$ which is less than $1$ for $n$ odd and greater than $1$ for $n$ even and has no limit as $n \to \infty$.

Consider $u_n$ such that $u_{2n}=\frac 1 {2n}$ and $u_{2n+1}=\frac 1 {\sqrt{2n}}$.

$u_n\rightarrow 0$ but $\frac {u_{2n+1}} {u_{2n}}=\sqrt{2n}\rightarrow\infty$