Gregory number - Alchetron, The Free Social Encyclopedia

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

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 .

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

π 4 = arctan ⁡ 1

is a Gregory number.

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