📝 题目
4. 1.21 设 ${x}_{1} > 0,{x}_{n + 1} = 1 + \frac{1}{{x}_{n}}\left( {n = 1,2,\cdots }\right)$ . 求证:
(1) $\displaystyle{1 \leq \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} \leq \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} \leq 2}$ ;
(2) $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n}}$ 存在,并求其极限值.
💡 答案与解析
4. 1.21 (2) $\frac{\sqrt{5} + 1}{2}$ .