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

${\displaystyle G_{x}=\sum _{i=0}^{\infty }(-1)^{i}{\frac {1}{(2i+1)x^{2i+1))))$

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

${\displaystyle G_{x}=\arctan {\frac {1}{x)).}$

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

${\displaystyle {\frac {\pi }{4))=\arctan 1}$

is a Gregory number.

Properties

• ${\displaystyle G_{-x}=-(G_{x})}$
• ${\displaystyle \tan(G_{x})={\frac {1}{x))}$