📝 题目
例 10 求证: 将级数 $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{\left( -1\right) }^{n - 1}}{\sqrt{n}}$ 重排后的级数
$$ 1 + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{{4k} - 3}} + \frac{1}{\sqrt{{4k} - 1}} - \frac{1}{\sqrt{2k}} + \cdots $$
发散.
💡 答案与解析
证 先考虑级数 $\mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {\frac{1}{\sqrt{{4k} - 3}} + \frac{1}{\sqrt{{4k} - 1}} - \frac{1}{\sqrt{2k}}}\right)$ ,因
$$ \frac{1}{\sqrt{{4k} - 3}} + \frac{1}{\sqrt{{4k} - 1}} - \frac{1}{\sqrt{2k}} \geq \frac{1}{\sqrt{4k}} + \frac{1}{\sqrt{4k}} - \frac{1}{\sqrt{2k}} $$
$$ = \left| {1 - \frac{1}{\sqrt{2}}}\right| \frac{1}{\sqrt{k}}, $$
及 $\displaystyle{\mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{1}{\sqrt{k}}}$ 发散,故 $\mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {\frac{1}{\sqrt{{4k} - 3}} + \frac{1}{\sqrt{{4k} - 1}} - \frac{1}{\sqrt{2k}}}\right)$ 发散 $\Rightarrow$ 重排后级数发散.