Gregory number

In mathematics, a Gregory number, named after James Gregory, is a real number of the form:

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

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

$G_x=\arctan \frac{1}{x}.$

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

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

is a Gregory number.

Properties

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

See also

• Størmer number