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|authorlink =John H. Conway|coauthors=R. K. Guy|title=The Book of Numbers|publisher=Copernicus Press|location=New York|year=1996|pages=241–243]

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

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

: $G\_x\; =\; arctanfrac\{1\}\{x\}.$

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

ee also

* Størmer number

References