ART

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

\( {\displaystyle G_{x}=\arctan {\frac {1}{x}}.} \)

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

\( {\displaystyle {\frac {\pi }{4}}=\arctan 1} \)

is a Gregory number.
See also

Størmer number

References

Conway, John H.; R. K. Guy (1996). The Book of Numbers. New York: Copernicus Press. pp. 241–243.

vte

Real numbers

0.999... Absolute difference Cantor set Cantor–Dedekind axiom Completeness Construction Decidability of first-order theories Extended real number line Gregory number Irrational number Normal number Rational number Rational zeta series Real coordinate space Real line Tarski axiomatization Vitali set

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License