1.1.3 和与积

1.1.3.1 和

使用求和号 $\sum$ 来缩写一个和

\begin{matrix} (1.8) & a_{1} + a_{2} + \dots + a_{n} = \sum_{k = 1}^{n} a_{k} . \end{matrix}

借助于这一符号可以表示 $n$ 个被加项 $a_{k} (k = 1, 2, \dots, n)$ 的和, $k$ 称为变动指标或求和变量.

(1) 相等各项 (即 $a_{k} = a, k = 1, 2, \dots, n$ ) 之和

\begin{matrix} (1.9a) & \sum_{k = 1}^{n} a_{k} = n a \end{matrix}

(2)各项与一常数相乘

\begin{matrix} (1.9b) & \sum_{k = 1}^{n} c a_{k} = c \sum_{k = 1}^{n} a_{k} . \end{matrix}

(3) 分部求和

\begin{matrix} (1.9c) & \sum_{k = 1}^{n} a_{k} = \sum_{k = 1}^{m} a_{k} + \sum_{k = m + 1}^{n} a_{k} (1 < m < n) . \end{matrix}

(4)相同长度的和相加

\begin{matrix} (1.9d) & \sum_{k = 1}^{n} (a_{k} + b_{k} + c_{k} + \dots) = \sum_{k = 1}^{n} a_{k} + \sum_{k = 1}^{n} b_{k} + \sum_{k = 1}^{n} c_{k} + \dots . \end{matrix}

(5) 重新编号

\begin{matrix} (1.9e) & \sum_{k = 1}^{n} a_{k} = \sum_{k = m}^{m + n - 1} a_{k - m + 1}, \sum_{k = m}^{n} a_{k} = \sum_{k = l}^{n - m + l} a_{k + m - l} . \end{matrix}

(6) 双重和中变换求和次序

\begin{matrix} (1.9f) & \sum_{i = 1}^{n} (\sum_{k = 1}^{m} a_{i k}) = \sum_{k = 1}^{m} (\sum_{i = 1}^{n} a_{i k}) . \end{matrix}

积的缩写记号是求积号 $\prod$ :

\begin{matrix} (1.10) & a_{1} a_{2} \dots a_{n} = \prod_{k = 1}^{n} a_{k} . \end{matrix}

借助于这一符号可以表示 $n$ 个因子 $a_{k} (k = 1, 2, \dots, n)$ 的积, $k$ 称为变动指标.

(1)重合因子 (即 $a_{k} = a, k = 1, 2, \dots, n$ ) 之积

\begin{matrix} (1.11a) & \prod_{k = 1}^{n} a_{k} = a^{n} \end{matrix}

(2)各因子乘以一常数

\begin{matrix} (1.11b) & \prod_{k = 1}^{n} (c a_{k}) = c^{n} \prod_{k = 1}^{n} a_{k} . \end{matrix}

(3) 分部求积

\begin{matrix} (1.11c) & \prod_{k = 1}^{n} a^{k} = (\prod_{k = 1}^{m} a_{k}) (\prod_{k = m + 1}^{n} a_{k}) (1 < m < n) . \end{matrix}

(4) 乘积的积

\begin{matrix} (1.11d) & \prod_{k = 1}^{n} a_{k} b_{k} c_{k} \dots = (\prod_{k = 1}^{n} a_{k}) (\prod_{k = 1}^{n} b_{k}) (\prod_{k = 1}^{n} c_{k}) \dots . \end{matrix}

(5) 重新编号

\begin{matrix} (1.11e) & \prod_{k = 1}^{n} a_{k} = \prod_{k = m}^{m + n - 1} a_{k - m + 1}, \prod_{k = m}^{n} a_{k} = \prod_{k = l}^{n - m + l} a_{k + m - l} . \end{matrix}

(6) 双重积中变换求积次序

\begin{matrix} (1.11f) & \prod_{i = 1}^{n} (\prod_{k = 1}^{m} a_{i k}) = \prod_{k = 1}^{m} (\prod_{i = 1}^{n} a_{i k}) . \end{matrix}