python/数学课程/有理数巧算之裂项相消.md

![](http://dsideal.obs.cn-north-1.myhuaweicloud.com/HuangHai/BlogImages/2023/02/ae96c6b41873b15860aab1f4160656f6.png)

答：$\large \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2018}-\frac{1}{2019}=\frac{2018}{2019}$

![](http://dsideal.obs.cn-north-1.myhuaweicloud.com/HuangHai/BlogImages/2023/02/1c8eef5eff05982f58e2046fc2d10b24.png)


$\because \large \frac{3}{4 \times 7}=\frac{1}{4}-\frac{1}{7}$

$\therefore \large  1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{46}-\frac{1}{56}=\frac{55}{56}$

**总结：**

![](http://dsideal.obs.cn-north-1.myhuaweicloud.com/HuangHai/BlogImages/2023/02/723ac3cfee3be299e551f5ccad09a0e9.png)