📝 题目
4.2.3 求证下列级数在所示区间上一致收敛:
(1) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\sin {nx}}{x + {2}^{n}}\;\left( {x > - 2}\right)$ ;
(2) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{x}^{2}{\mathrm{e}}^{-{nx}}\;\left( {x \geq 0}\right)$ ;
(3) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{n}^{2}}{\sqrt{n!}}\left( {{x}^{n} + {x}^{-n}}\right) \;\left( {\frac{1}{2} \leq \left| x\right| \leq 2}\right)$ ;
(4) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{x}^{n}{\ln }^{n}x\;\left( {0 \leq x \leq 1}\right)$ .