Question

问题

Questions in category: 极限 (Limit).

设数列 ${x_{n}}_{n = 1}^{\infty}$ 满足 $x_{n + 1} = 1 + \frac{x_{n}}{1 + x_{n}}$ , $x_{1} = 1$ , 证明 $lim_{n \to + \infty} x_{n}$ 存在, 并求此极限.

Posted by haifeng on 2017-12-18 16:48:13 last update 2017-12-18 20:54:55 | Answers (1) | 收藏

设 $x_{1} = 1$ , $x_{2} = 1 + \frac{x_{1}}{1 + x_{1}}$ , $\dots$ , $x_{n} = 1 + \frac{x_{n - 1}}{1 + x_{n - 1}}$ , $n = 2, 3, \dots$ .

我们写出这个数列 ${x_{n}}_{n = 1}^{\infty}$ 的前几项:

$\frac{1}{1}, \frac{3}{2}, \frac{8}{5}, \frac{21}{13}, \frac{55}{34}, \frac{144}{89}, \dots$

证明:

(1) 如果 $x_{n}$ 表示为既约分数 $\frac{a_{n}}{b_{n}}$ , 则有

$b_{n}^{2} = a_{n} \cdot a_{n - 1} + 1, b_{n} \cdot b_{n + 1} = a_{n}^{2} + 1.$

(2) 极限 $lim_{n \to + \infty} x_{n}$ 存在, 并求此极限值.