📝 题目
4. 3.1 求下列幂级数的收敛半径,并讨论收敛区间端点的收敛性:
(1) $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{1 + \frac{1}{2} + \cdots + \frac{1}{n}}{n}{x}^{n}}$ ; (2) $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\left( {2n}\right) !!}{\left( {{2n} + 1}\right) !!}{x}^{n}$ ;
(3) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( 1 + \frac{1}{n}\right) }^{{n}^{2}}{x}^{2n}$ ; (4) $\displaystyle{\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{2}^{n} + {3}^{n}}{n}{x}^{n}}$ ;
(5) $\mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( 1 + 2\cos \frac{n\pi }{4}\right) }^{n}{x}^{n}$ .
💡 答案与解析
4. 3.1 (1) $R = 1$ ,当 $x = - 1$ 时收敛,当 $x = 1$ 时发散;
(2) $R = 1$ ,当 $x = - 1$ 时收敛,当 $x = 1$ 时发散；
(3) $R = \frac{1}{\sqrt{\mathrm{e}}}$ ,当 $x = \pm \frac{1}{\sqrt{\mathrm{e}}}$ 时发散；
(4) $R = \frac{1}{3}$ ,当 $x = - \frac{1}{3}$ 时收敛,当 $x = \frac{1}{3}$ 时发散;
(5) $R = \frac{1}{3}$ ,当 $x = \pm \frac{1}{3}$ 时发散.