Gregory number

In mathematics, a Gregory number, named after James Gregory, is a real number of the form:{{cite book|first=John H.|last=Conway|author-link =John H. Conway|author2=R. K. Guy|author2-link=R. K. Guy|title=The Book of Numbers|url=https://archive.org/details/bookofnumbers0000conw|url-access=registration|publisher=Copernicus Press|location=New York|year=1996|pages=[https://archive.org/details/bookofnumbers0000conw/page/241 241–243]}}

: $G\_x\; =\; \backslash sum\_\{i\; =\; 0\}^\backslash infty\; (-1)^i\; \backslash frac\{1\}\{(2i\; +\; 1)x^\{2i\; +\; 1\}\}$

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

: $G\_x\; =\; \backslash arctan\backslash frac\{1\}\{x\}.$

Setting x = 1 gives the well-known Leibniz formula for pi. Thus, in particular,

:$\backslash frac\{\backslash pi\}\{4\}=\backslash arctan\; 1$

is a Gregory number.