Gregory number

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

[math]\displaystyle{ G_x = \sum_{i = 0}^\infty (-1)^i \frac{1}{(2i + 1)x^{2i + 1}} }[/math]

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

[math]\displaystyle{ G_x = \arctan\frac{1}{x}. }[/math]

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

[math]\displaystyle{ \frac{\pi}{4}=\arctan 1 }[/math]

is a Gregory number.

Properties

• [math]\displaystyle{ G_{-x}=-(G_x) }[/math]
• [math]\displaystyle{ \tan(G_x)= \frac{1}{x} }[/math]

See also

• Størmer number