Gregory number

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

where x is any rational number greater or equal to 1. Considering the

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

Setting x = 1 gives the well-known

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

is a Gregory number.

Properties

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

See also