📝 题目
4.1.13 求证: 若级数 $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}^{2}}$ 及 $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }{b}_{n}^{2}}$ 收敛,则下面级数:
$$ \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \cdot {b}_{n},\;\mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( {a}_{n} + {b}_{n}\right) }^{2},\;\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{n} $$
皆收敛.
💡 答案与解析
4. 1.12 (1) $\left\{ \begin{array}{l} \text{ 当 }p > 1\text{ 时绝对收敛, } \\ \text{ 当 }\frac{1}{2} < p < 1\text{ 时条件收敛; } \end{array}\right.$ (2) $\left\{ \begin{array}{l} \text{ 当 }p > 1\text{ 时绝对收敛, } \\ \text{ 当 }0 < p \leq 1\text{ 时条件收敛 } \end{array}\right.$