The series for the inverse tangent, tan^(-1) x = x - 1/3 x^3 + 1/5 x^5 + .... Plugging in x = 1 gives Gregory's formula 1/4 π = 1 - 1/3 + 1/5 - 1/7 + 1/9 - .... This series is intimately connected with the number of representations of n by k squares r_k(n), and also with Gauss's circle problem (Hilbert and Cohn-Vossen 1999, pp. 27-39).

Gauss's circle problem | Gregory series | inverse tangent | pi formulas | sum of squares function