Leibniz Series

The series for the inverse tangent,


Plugging in x=1 gives Gregory's formula


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).

See also

Gauss's Circle Problem, Gregory Series, Inverse Tangent, Pi Formulas, Sum of Squares Function

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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.

Referenced on Wolfram|Alpha

Leibniz Series

Cite this as:

Weisstein, Eric W. "Leibniz Series." From MathWorld--A Wolfram Web Resource.

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