Chudnovsky series (${\pi }^{-1}=\frac{12}{640320^{3/2}}{\sum }_{k\ge 0}\frac{(-1{)}^k(6k)!(13591409+545140134k)}{(3k)!(k!{)}^{3}640320^{3k}}$)

The Chudnovsky series is the rapidly converging identityThe absolute ratio of successive summands tends to $1728/640320^3$, approximately $6.58\times 10^{-15}$, giving about fourteen new decimal digits per term asymptotically. Keeping only $k=0$ gives $\pi \approx 640320^{3/2}/(12\cdot 13591409)$. The factorial ratio, not a direct geometric approximation of a circle, controls the numerical convergence.^[ref]