代數/本書課文/求和/阿貝爾變換

阿貝爾變換，也叫分部求和法，把兩組數列乘積的和表達成一組數列和與一組數列差。

$\begin{matrix} \sum_{i = 1}^{n} a_{i} b_{i} & = a_{1} (b_{1} - b_{2}) + (a_{1} + a_{2}) (b_{2} - b_{3}) + \dots \\ + (a_{1} + a_{2} + \dots + a_{n - 1}) (b_{n - 1} - b_{n}) + (a_{1} + a_{2} + \dots + a_{n}) b_{n} \end{matrix}$

例子：求

\sum_{k = 1}^{n} k q^{k - 1}

$\begin{matrix} \sum_{k = 1}^{n} k q^{k - 1} & = - (1) - (1 + q) - \dots - (1 + q + \dots + q^{n - 1}) + n (1 + q + \dots + q^{n}) \\ = - \frac{1 - q}{1 - q} - \frac{1 - q^{2}}{1 - q} - \dots - \frac{1 - q^{n}}{1 - q} + n \frac{1 - q^{n + 1}}{1 - q} \\ = - \frac{n - 1 - q - q^{2} - \dots - q^{n}}{1 - q} + n \frac{1 - q^{n + 1}}{1 - q} \\ = \frac{1 - q^{n + 1}}{(1 - q)^{2}} - \frac{n q^{n + 1}}{1 - q} = \frac{1 - (n + 1) q^{n + 1} + n q^{n + 2}}{(1 - q)^{2}} \end{matrix}$