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4.md
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......@@ -321,15 +321,13 @@ array([ 1.5, 1. , 0.5, 0. , -0.5, -1. , -1.5])
伟大的德国数学家和哲学家戈特弗里德·威廉·莱布尼茨（Gottfried Wilhelm Leibniz，1646 ~ 1716年）发现了一个简单分数的无穷和。 公式是：
![](img/4-4.svg)
![](http://latex.codecogs.com/gif.latex?%5Cpi%20%3D%202%20%5Ccdot%20%5Cleft%28%20%5Cfrac%7B2%7D%7B1%7D%5Ccdot%5Cfrac%7B2%7D%7B3%7D%5Ccdot%5Cfrac%7B4%7D%7B3%7D%5Ccdot%5Cfrac%7B4%7D%7B5%7D%5Ccdot%5Cfrac%7B6%7D%7B5%7D%5Ccdot%5Cfrac%7B6%7D%7B7%7D%5Cdots%20%5Cright%29)
虽然需要一些数学来确定它，但我们可以用数组来说服我们自己，公式是有效的。 让我们计算莱布尼茨的无穷和的前 5000 个项，看它是否接近 π。
![](img/4-5.svg)
我们将计算这个有限的总和，首先加上所有的正项，然后减去所有负项的和 [1]：
![](img/4-6.svg)
![](http://latex.codecogs.com/gif.latex?%5Cpi%20%5Capprox%202%20%5Ccdot%20%5Cleft%28%20%5Cfrac%7B2%7D%7B1%7D%20%5Ccdot%20%5Cfrac%7B4%7D%7B3%7D%20%5Ccdot%20%5Cfrac%7B6%7D%7B5%7D%20%5Ccdots%20%5Cfrac%7B1%2C000%2C000%7D%7B999999%7D%20%5Cright%29%20%5Ccdot%20%5Cleft%28%20%5Cfrac%7B2%7D%7B3%7D%20%5Ccdot%20%5Cfrac%7B4%7D%7B5%7D%20%5Ccdot%20%5Cfrac%7B6%7D%7B7%7D%20%5Ccdots%20%5Cfrac%7B1%2C000%2C000%7D%7B1%2C000%2C001%7D%20%5Cright%29)
> [1] 令人惊讶的是，当我们将无限多个分数相加时，顺序可能很重要。但是我们对 π 的近似只使用了大量的数量有限的分数，所以可以按照任何方便的顺序，将这些项相加。
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