How can I make a sequence $u_n$ so that $u_{2n+1}$ is increasing and $u_{2n}$ is decreasing?

Question

I know that, the sequence $(u_n)$ so that $u_1 = 1$, $u_{n+1} = 1 + \dfrac{1}{u_n}, \forall n \geqslant 1$ has properties $u_{2n+1}$ is increasing and $u_{2n}$ is decreasing. Now I want to construct more sequences like that. I don't know how to start. How can I construct?

Answer 1

Let $a_n$ be any increasing sequence and $b_n$ any decreasing sequence. Then $$u_n=(1+(-1)^n)a_n+(1-(-1)^n)b_n$$ has the property you want.

Answer 2

One example is $$u_n = (-1)^n \frac{1}{n}$$