📝 题目
4. 1.10 判断下列级数的收敛性:
(1) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n}\frac{{\sin }^{2}n}{n}$ ; (2) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{\left( -1\right) }^{n - 1}}{n} \cdot \frac{{a}^{n}}{1 + {a}^{n}}\left( {a > 0}\right)$ ;
(3) $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\sin n \cdot \sin {n}^{2}}{n}}$ ; (4) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\sin \left( {n + \frac{1}{n}}\right) }{n}$ .
💡 答案与解析
4. 1.8 利用上题结论,设 ${a}_{n}\frac{\text{ 定义 }}{}\frac{n}{{p}_{1} + {p}_{2} + \cdots + {p}_{n}}$ ,可证 ${a}_{2n} \leq \frac{2}{{p}_{n}}$ . 4.1.9 (1) 收敛；(2)收敛；(3)收敛；(4)收敛. 4.1.10 (1) 收敛；(2)收敛；(3)收敛；(4)收敛.
4.1.11 (1) $\left\{ \begin{array}{l} \text{ 当 }\left| x\right| \neq 1\text{ 时绝对收敛 } \\ \text{ 当 }\left| x\right| = 1\text{ 时发散; } \end{array}\right.$ (2) $\left\{ \begin{array}{l} \text{ 当 }x \geq 0\text{ 时绝对收敛, } \\ \text{ 当 }x < 0\text{ 时发散. } \end{array}\right.$