Gregory number

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. Thus, in particular,

{\frac {\pi }{4}}=\arctan 1

is a Gregory number.

References

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

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