Leibniz Series

The series for the inverse tangent,

Plugging in  gives Gregory's formula

This series is intimately connected with the number of representations of  by  squares , and also with Gauss's circle problem (Hilbert and Cohn-Vossen 1999, pp. 27-39).

REFERENCES:

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 37, 1999.

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 50, 1986.