Gregory number explained

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

G _{x} = \sum ^{inf ty} _{i =0} ( -1 ) ^{i} (1)/(( 2 i +1 )x ^{2i})

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

G _{x} = \arctan (1)/(x) .

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

(\pi)/(4) = \arctan 1 is a Gregory number.

Properties

G _{-x} =- (G _{x)} \tan(G _{x)=} (1)/(x)

See also

• Størmer number

Notes and References

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