问题

用来计算 $\pi$ 的楚德诺夫斯基(Chudnovsky)公式:

\[
\frac{1}{\pi}=12\sum_{k=0}^{+\infty}\frac{(-1)^k\cdot(6k)!(13591409+545140134k)}{(3k)!(k!)^3\cdot 640320^{3k+\frac{3}{2}}}.
\]

等价于

\[
\frac{(640320)^{\frac{3}{2}}}{12\pi}=\frac{426880\sqrt{10005}}{\pi}=\sum_{k=0}^{+\infty}\frac{(6k)!(545140134k+13591409)}{(3k)!(k!)^3(-262537412640768000)^k}
\]

\[
\frac{640320^{3/2}}{12\pi}=\frac{426880\sqrt{10005}}{\pi}=\sum^\infty_{k=0}\frac{(6k)!(545140134k+13591409)}{(3k)!(k!)^3\left(-640320\right)^{3k}}
\]

References:

https://en.wikipedia.org/wiki/Chudnovsky_algorithm

https://bbs.emath.ac.cn/thread-17586-1-1.html

http://numbers.computation.free.fr/Constants/PiProgram/pifast.html

http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#pi_chudnovsky