📝 题目
1.3.3 设 $c > 1$ ,求序列 $\sqrt{c},\sqrt{c\sqrt{c}},\sqrt{c\sqrt{c\sqrt{c}}},\cdots$ 的极限.
💡 答案与解析
1.3.3 对给定的 $\displaystyle{c > 1,{x}_{n + 1} = \sqrt{c{x}_{n}} \Rightarrow {x}_{n} \uparrow ,0 < {x}_{n} < c \Rightarrow \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = c}$ 或
$$ \frac{{x}_{n + 1}}{c} = \sqrt{\frac{{x}_{n}}{c}}\overset{{y}_{n} = \frac{{x}_{n}}{c}}{ = }{y}_{n + 1} = \sqrt{{y}_{n}} $$
$$ \Rightarrow {y}_{n} = {\left( {y}_{n - 1}\right) }^{\frac{1}{2}} = \cdots = {\left( {y}_{1}\right) }^{\frac{1}{{2}^{n - 1}}} \rightarrow 1 \Rightarrow \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = c. $$