Gregory number

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


 * $$G_x = \sum_{i = 0}^\infty (-1)^i \frac{1}{(2i + 1)x^{2i + 1}}$$

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


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

Setting x = 1 gives the well-known Leibniz formula for pi. Thus, in particular,
 * $$\frac{\pi}{4}=\arctan 1$$

is a Gregory number.

Properties

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