📝 题目
4.1.2 判断下列级数的收敛性:
(1) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{n}^{2}}{{\left( 1 + \frac{1}{n}\right) }^{{n}^{2}}}$ ; (2) $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\ln n}{{2}^{n}}}$ ;
(3) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{n}^{n - 1}}{{\left( 2{n}^{2} + 1\right) }^{n/2}}$ ; (4) $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}^{n}}{n!}}$ ;
(5) $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\sqrt{n!}}{{n}^{n/2}}}$ ; (6) $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\sqrt{n!} \cdot {2}^{n}}{{n}^{n/2}}}$ ;
(7) $\frac{3}{1} + \frac{3 \cdot 5}{1 \cdot 4} + \frac{3 \cdot 5 \cdot 7}{1 \cdot 4 \cdot 7} + \frac{3 \cdot 5 \cdot 7 \cdot 9}{1 \cdot 4 \cdot 7 \cdot {10}} + \cdots$ .
💡 答案与解析
4.1.2 (1) 收敛；(2)收敛；(3)收敛；(4)收敛；(5)收敛；
(6)发散；(7)收敛.