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.
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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
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