The Gregory series is a pi formula found by Gregory and Leibniz and obtained by plugging x = 1 into the Leibniz series, π/4 = sum_(k = 1)^∞ (-1)^(k + 1)/(2k - 1) = 1 - 1/3 + 1/5 - ... (Wells 1986, p. 50). The formula converges very slowly, but its convergence can be accelerated using certain transformations, in particular π = sum_(k = 1)^∞ (3^k - 1)/4^k ζ(k + 1), where ζ(z) is the Riemann zeta function. Taking the partial series gives the analytic result 4 sum_(k = 1)^N (-1)^(k + 1)/(2k - 1) = π + (-1)^N[ψ_0(1/4 + 1/2 N) - ψ_0(3/4 + 1/2 N)].

Gregory's formula | Leibniz series | Machin-like formulas | Machin's formula | Mercator series | pi formulas