7.2.4 某些特殊级数

7.2.4.1 一些重要数项级数的值

\begin{matrix} (7.39) & 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots + \frac{1}{n!} + \dots = e \end{matrix}

\begin{matrix} (7.40) & 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{{(- 1)}^{n}}{n!} + \dots = \frac{1}{e}, \end{matrix}

\begin{matrix} (7.41) & 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots + \frac{{(- 1)}^{n + 1}}{n} + \dots = \ln 2, \end{matrix}

\begin{matrix} (7.42) & 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^{n}} + \dots = 2 \end{matrix}

\begin{matrix} (7.43) & 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \dots + \frac{{(- 1)}^{n}}{2^{n}} + \dots = \frac{2}{3}, \end{matrix}

\begin{matrix} (7.44) & 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots + \frac{{(- 1)}^{n - 1}}{2 n - 1} + \dots = \frac{π}{4}, \end{matrix}

\begin{matrix} (7.45) & \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{n (n + 1)} + \dots = 1, \end{matrix}

\begin{matrix} (7.46) & \frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \dots + \frac{1}{(2 n - 1) (2 n + 1)} + \dots = \frac{1}{2}, \end{matrix}

\begin{matrix} (7.47) & \frac{1}{1 \cdot 3} + \frac{1}{2 \cdot 4} + \frac{1}{3 \cdot 5} + \dots + \frac{1}{(n - 1) (n + 1)} + \dots = \frac{3}{4}, \end{matrix}

\begin{matrix} (7.48) & \frac{1}{3 \cdot 5} + \frac{1}{7 \cdot 9} + \frac{1}{11 \cdot 13} + \dots + \frac{1}{(4 n - 1) (4 n + 1)} + \dots = \frac{1}{2} - \frac{π}{8}, \end{matrix}

\begin{matrix} (7.49) & \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \dots + \frac{1}{n (n + 1) (n + 2)} + \dots = \frac{1}{4}, \end{matrix}

\frac{1}{1 \cdot 2 \cdot \dots \cdot l} + \frac{1}{2 \cdot 3 \cdot \dots \cdot (l + 1)} + \dots + \frac{1}{n \cdot \dots \cdot (n + l - 1)} + \dots = \frac{1}{(l - 1) (l - 1)!},

(7.50)

\begin{matrix} (7.51) & 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \dots + \frac{1}{n^{2}} + \dots = \frac{π^{2}}{6}, \end{matrix}

\begin{matrix} (7.52) & 1 - \frac{1}{2^{2}} + \frac{1}{3^{2}} - \frac{1}{4^{2}} + \dots + \frac{{(- 1)}^{n + 1}}{n^{2}} + \dots = \frac{π^{2}}{12}, \end{matrix}

\begin{matrix} (7.53) & \frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + \dots + \frac{1}{{(2 n + 1)}^{2}} + \dots = \frac{π^{2}}{8}, \end{matrix}

\begin{matrix} (7.54) & 1 + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \frac{1}{4^{4}} + \dots + \frac{1}{n^{4}} + \dots = \frac{π^{4}}{90}, \end{matrix}

\begin{matrix} (7.55) & 1 - \frac{1}{2^{4}} + \frac{1}{3^{4}} - \dots + \frac{{(- 1)}^{n + 1}}{n^{4}} + \dots = \frac{7 π^{4}}{720}, \end{matrix}

\begin{matrix} (7.56) & \frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + \dots + \frac{1}{{(2 n + 1)}^{4}} + \dots = \frac{π^{4}}{96}, \end{matrix}

\begin{matrix} (7.57) & 1 + \frac{1}{2^{2 k}} + \frac{1}{3^{2 k}} + \frac{1}{4^{2 k}} + \dots + \frac{1}{n^{2 k}} + \dots = \frac{π^{2 k} 2^{2 k - 1}}{(2 k)!} B_{k}^{(1)}, \end{matrix}

\begin{matrix} (7.58) & 1 - \frac{1}{2^{2 k}} + \frac{1}{3^{2 k}} - \frac{1}{4^{2 k}} + \dots + \frac{{(- 1)}^{n + 1}}{n^{2 k}} + \dots = \frac{π^{2 k} (2^{2 k - 1} - 1)}{(2 k)!} B_{k}, \end{matrix}

\begin{matrix} (7.59) & 1 + \frac{1}{3^{2 k}} + \frac{1}{5^{2 k}} + \frac{1}{7^{2 k}} + \dots + \frac{1}{{(2 n - 1)}^{2 k}} + \dots = \frac{π^{2 k} (2^{2 k} - 1)}{2 \cdot (2 k)!} B_{k}, \end{matrix}

\begin{matrix} (7.60) & 1 - \frac{1}{3^{2 k + 1}} + \frac{1}{5^{2 k + 1}} - \frac{1}{7^{2 k + 1}} + \dots + \frac{{(- 1)}^{n + 1}}{{(2 n - 1)}^{2 k + 1}} + \dots = \frac{π^{2 k + 1}}{2^{2 k + 2} (2 k)!} E_{k}^{(2)} . \end{matrix}

① $B_{k}$ 是伯努利数.

② $E_{k}$ 是欧拉数.

(1) 伯努利数的第一定义 某些特殊函数的幂级数展开式中会出现伯努利数 $B_{k}$ ,如三角函数 $\tan x, \cot x, \csc x$ ,以及双曲函数 $\tanh x, \coth x, \cos ech x$ . 伯努利数 $B_{k}$ 可定义如下:

\begin{matrix} (7.61) & \frac{x}{e^{x} - 1} = 1 - \frac{x}{2} + B_{1} \frac{x^{2}}{2!} - B_{2} \frac{x^{4}}{4!} \pm \dots + {(- 1)}^{n + 1} B_{n} \frac{x^{2 n}}{(2 n)!} \pm \dots (| x | < 2 π) \end{matrix}

利用系数对比法将其与 $x$ 的方幂作比较,可计算伯努利数. 用此方法计算的前几个伯努利数见表 7.1.

$k$	$B_{k}$	$k$	$B_{k}$	$k$	$B_{k}$	$k$	$B_{k}$
1	$\frac{1}{6}$	4		7	$\frac{7}{6}$	10	$\frac{174611}{330}$
2	$\frac{1}{30}$	5	$\begin{array}{r} \frac{1}{30} \\ \frac{5}{66} \\ 691 \\ 773 \end{array}$	8	3617	11	854513 138
3	$\frac{1}{42}$	6		9	43867 798

(2) 伯努利数的第二定义 有些人把伯努利数定义如下:

\begin{matrix} (7.62) & \frac{x}{e^{x} - 1} = 1 + \overset{―}{B_{1}} \frac{x}{1!} + \overset{―}{B_{2}} \frac{x^{2}}{2!} + \dots + \overset{―}{B_{2 n}} \frac{x^{2 n}}{(2 n)!} + \dots (| x | < 2 π) . \end{matrix}

由此得到递归公式

\begin{matrix} (7.63) & \overset{―}{B_{k + 1}} = {(\bar{B} + 1)}^{k + 1} (k = 1, 2, 3, \dots), \end{matrix}

利用二项式定理 (参见第 14 页 1.1.6.4,1.),用 $\overset{―}{B_{v}}$ 代替 ${\bar{B}}^{v}$ ,即指数变为下标. 前几个新数为

\overset{―}{B_{1}} = - \frac{1}{2}, \overset{―}{B_{2}} = \frac{1}{6}, \overset{―}{B_{4}} = - \frac{1}{30}, \overset{―}{B_{6}} = \frac{1}{42},

\begin{matrix} (7.64) & \overset{―}{B_{8}} = - \frac{1}{30}, \overset{―}{B_{10}} = \frac{5}{66}, \overset{―}{B_{12}} = - \frac{691}{2730}, \overset{―}{B_{14}} = \frac{7}{6}, \end{matrix}

\overset{―}{B_{16}} = - \frac{3617}{510}, \dots, \overset{―}{B_{3}} = \overset{―}{B_{5}} = \overset{―}{B_{7}} = \dots = 0,

且有

\begin{matrix} (7.65) & B_{k} = {(- 1)}^{k + 1} \overset{―}{B_{2 k}} (k = 1, 2, 3, \dots) . \end{matrix}

(3) 欧拉数的第一定义 某些特殊函数的幂级数展开式中会出现欧拉数 $E_{k}$ ,如函数 $\sec x$ 和 $sech x$ . 欧拉数可定义如下:

\begin{matrix} (7.66) & \sec x = 1 + E_{1} \frac{x^{2}}{2!} + E_{2} \frac{x^{4}}{4!} + \dots + E_{n} \frac{x^{2 n}}{(2 n)!} + \dots (| x | < \frac{π}{2}) . \end{matrix}

利用系数对比法将其与 $x$ 的方幂作比较,可计算欧拉数. 表 7.2 列出了一些欧拉数的值.

(4) 欧拉数的第二定义 与 (7.63) 类似, 欧拉数也可用递归公式来定义:

\begin{matrix} (7.67) & {(\bar{E} + 1)}^{k} + {(\bar{E} - 1)}^{k} = 0 (k = 1, 2, 3, \dots) . \end{matrix}

利用二项式定理,用 $\overset{―}{E_{v}}$ 代替 ${\bar{E}}^{v}$ ,可得到前几个欧拉数的值:

\overset{―}{E_{2}} = - 1, \overset{―}{E_{4}} = 5, \overset{―}{E_{6}} = - 61, \overset{―}{E_{8}} = 1385,

\begin{matrix} (7.68) & \overset{―}{E_{10}} = - 50521, \overset{―}{E_{12}} = 2702765, \overset{―}{E_{14}} = - 199360981, \end{matrix}

\overset{―}{E_{16}} = 19391512145, \dots, \overset{―}{E_{1}} = \overset{―}{E_{3}} = \overset{―}{E_{5}} = \dots = 0,

且有

\begin{matrix} (7.69) & E_{k} = {(- 1)}^{k} \overset{―}{E_{2 k}} (k = 1, 2, 3, \dots) . \end{matrix}

(5) 欧拉数与伯努利数间的关系 欧拉数与伯努利之间的关系为

\begin{matrix} (7.70) & \overset{―}{E_{2 k}} = \frac{4^{2 k + 1}}{2 k + 1} {(\overset{―}{B_{k}} - \frac{1}{4})}^{2 k + 1} (k = 1, 2, \dots) . \end{matrix}