In mathematics, a Gregory number, named after James Gregory, is a real number of the form:
G x = ∑ i = 0 ∞ ( − 1 ) i 1 ( 2 i + 1 ) x 2 i + 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
G x = arctan 1 x . {\displaystyle G_{x}=\arctan {\frac {1}{x}}.}
Setting x = 1 gives the well-known Leibniz formula for pi. Thus, in particular,
π 4 = arctan 1 {\displaystyle {\frac {\pi }{4}}=\arctan 1}
is a Gregory number.
Properties
G − x = − ( G x ) {\displaystyle G_{-x}=-(G_{x})}
tan ( G x ) = 1 x {\displaystyle \tan(G_{x})={\frac {1}{x}}}
See also Størmer number
References