Gregory number

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In mathematics, a Gregory number, named after James Gregory, is a real number of the form:[1]

G_x = \sum_{i = 0}^\infty (-1)^i \frac{1}{(2i + 1)x^{2i + 1}}

where x is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we have

G_x = \arctan\frac{1}{x}.

Setting x = 1 gives the well-known Leibniz formula for pi.

[edit] See also

[edit] References

  1. ^ Conway, John H.; R. K. Guy (1996). The Book of Numbers. New York: Copernicus Press, 241–243. 
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