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Leibniz Series


The series for the inverse tangent,

 tan^(-1)x=x-1/3x^3+1/5x^5+....

Plugging in x=1 gives Gregory's formula

 1/4pi=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).


See also

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

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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.

Referenced on Wolfram|Alpha

Leibniz Series

Cite this as:

Weisstein, Eric W. "Leibniz Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeibnizSeries.html

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