📝 题目
4.1.9 判断下列级数的收敛性:
(1) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}\frac{{\ln }^{2}n}{n}$ ; (2) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}\frac{\sqrt{n}}{n + {100}}$ ；
(3) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}\frac{1 + \frac{1}{2} + \cdots + \frac{1}{n}}{n}$ ;
(4) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\sin \left( {\pi \sqrt{{n}^{2} + 1}}\right)$ .