Gregory's Formula

A series Formula for Pi found by Gregory and Leibniz,

$${\pi\over 4}=1-{1\over 3}+{1\over 5}+\ldots.$$

It converges very slowly, but its convergence can be accelerated using certain transformations, in particular

$$\pi=\sum_{k=1}^\infty {3^k-1\over 4^k}\zeta(k+1),$$

where $\zeta(z)$ is the Riemann Zeta Function (Vardi 1991).

See also Machin's Formula, Machin-Like Formulas, Pi

References

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 157-158, 1991.