📝 题目
例 10 设 ${x}_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$ ,求证: $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = + \infty}$ .
💡 答案与解析
证 显然 ${x}_{n}$ 是单调增加的,只要证明它不收敛即可. 对 ${\varepsilon }_{0} = \frac{1}{2}$ , 因为
$$ {x}_{2n} - {x}_{n} = \frac{1}{n + 1} + \frac{1}{n + 2} + \cdots + \frac{1}{2n} $$
$$ \geq \overset{n\text{ 项 }}{\overbrace{\frac{1}{2n} + \frac{1}{2n} + \cdots + \frac{1}{2n}}} = \frac{1}{2} = {\varepsilon }_{0}, $$
由收敛原理知 $\left\{ {x}_{n}\right\}$ 不收敛.