📝 题目
1.3.5 设 ${F}_{0} = {F}_{1} = 1,{F}_{n + 1} = {F}_{n} + {F}_{n - 1}$ ,求证: $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{F}_{n - 1}}{{F}_{n}} = \frac{\sqrt{5} - 1}{2}}$ .
💡 答案与解析
1.3.5 ${x}_{n}\overset{\text{ 定义 }}{ \approx }\frac{{F}_{n - 1}}{{F}_{n}} \Rightarrow {x}_{n + 1} = \frac{{F}_{n}}{{F}_{n + 1}} = \frac{{F}_{n}}{{F}_{n} + {F}_{n - 1}} = \frac{1}{1 + \frac{{F}_{n - 1}}{{F}_{n}}} = \frac{1}{1 + {x}_{n}}$ .