不用三角函数和积分可以写出 π 的表达式吗? - 知乎日报
知乎日报
![](http://picx.zhimg.com/v2-bce90bc98c4013bfd1c2b459bb9d7e1f_fhd.jpg?source=8673f162)
1.Francois Viete
![](https://www.zhihu.com/equation?tex=%5Cpi%3D2%5Ccdot%5Cfrac%7B2%7D%7B%5Csqrt%7B2%7D%7D%5Ccdot%5Cfrac%7B2%7D%7B%5Csqrt%7B2%2B%5Csqrt%7B2%7D%7D%7D%5Ccdot%5Cfrac%7B2%7D%7B%5Csqrt%7B2%2B%5Csqrt%7B2%2B%5Csqrt%7B2%7D%7D%7D%7D%5Ccdots%5C%5C)
2.John Wallis
![](https://www.zhihu.com/equation?tex=%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cprod_%7Bn%3D1%7D%5E%5Cinfty%5Cleft%28+%5Cfrac%7B%282n%29%5E2%7D%7B%282n-1%29%282n%2B1%29%7D+%5Cright%29%5C%5C)
3.Srinivasa Ramanujan
![](https://www.zhihu.com/equation?tex=%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7B2%7D%7B1%2B%5Cfrac%7B3%7D%7B1%2B%5Cfrac%7B4%7D%7B1%2B%5Ccdots%7D%7D%7D%7D%3D%5Csqrt%7B%5Cfrac%7B%5Cpi+e%7D%7B2%7D%7D-%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7B1%7D%7B%282n%2B1%29%21%21%7D%5C%5C)
![](https://www.zhihu.com/equation?tex=%5Cfrac%7B1%7D%7B%5Cpi%7D%3D%5Cfrac%7B1%7D%7B8%7D%5Csum_%7Bk%3D0%7D%5E%5Cinfty%2820k%2B3%29%5Cfrac%7B%28-1%29%5Ek%284k%29%21%7D%7B%284%5Csqrt%7B2%7D%29%5E%7B4k%7D%28k%21%29%5E4%7D%5C%5C)
4.Stirling
![](https://www.zhihu.com/equation?tex=%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7Be%5Enn%21%7D%7Bn%5En%5Csqrt%7Bn%7D%7D%3D%5Csqrt%7B2%5Cpi%7D%5C%5C)
5.Euler
![](https://www.zhihu.com/equation?tex=%5Cpi%3D%5Cfrac%7B2%7D%7B%5Cprod%5Climits_%7Bn%3D2%7D%5E%7B%5Cinfty%7D%5Cleft%28+1%2B%5Cfrac%7B%28-1%29%28p_n-1%29%7D%7B2p_n%7D+%5Cright%29%7D%5C%5C)
其中
![](https://www.zhihu.com/equation?tex=p_n)
是第
![](https://www.zhihu.com/equation?tex=n)
个素数.
Euler 解决的 Basel 问题
![](https://www.zhihu.com/equation?tex=%5Csum_%7Bk%3D1%7D%5E%5Cinfty%5Cfrac%7B1%7D%7Bk%5E2%7D%3D%5Cfrac%7B1%7D%7B1%5E2%7D%2B%5Cfrac%7B1%7D%7B2%5E2%7D%2B%5Ccdots%3D%5Cfrac%7B%5Cpi%5E2%7D%7B6%7D%5C%5C)
6.Leibniz
![](https://www.zhihu.com/equation?tex=%5Cfrac%7B%5Cpi%7D%7B4%7D%3D1-%5Cfrac%7B1%7D%7B3%7D%2B%5Cfrac%7B1%7D%7B5%7D-%5Cfrac%7B1%7D%7B7%7D%2B%5Ccdots%3D%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7D%28-1%29%5E%7Bk%2B1%7D%5Cfrac%7B1%7D%7B2k-1%7D%5C%5C)
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