<s>[INST] Compute limit of sequence Let $(x_n)$ be real sequences such that $x_{1}=\dfrac{1}{3}, x_{2n}=\dfrac{1}{3}x_{2n-1}, x_{2n+1}=\dfrac{1}{3}+x_{2n}, n=1,2,\cdots $.
Compute $$\lim_{x \to \infty} \sup x_{n} \text{ and } \lim_{x \to \infty} \inf x_{n}. $$
 [/INST] $$
x_{2n}=x_{2n-1}/3=(1/3+x_{2n-2})/3=x_{2n-2}/3+1/9
$$
$$
x{2n+1}=1/3+x_{2n}=1/3+x_{2n-1}/3
$$
and we have $x_1=1/3$, we can add $x_0=0$, which is ok by the definition
then sovle the recursive sequence, we get
$$
x_{2n} = \frac{1}{6}(1-\frac{1}{3^n})
$$
$$
x_{2n+1} = \frac{1}{2} - \frac{1}{6}\frac{1}{3^n}
$$
so
$$
\lim_{n \to \infty} \sup x_{n} = 1/2
$$
$$
\lim_{n \to \infty} \inf x_{n} = 1/6
$$
</s>