OFFSET
1,2
COMMENTS
This constant is formed from the continued fraction [1; 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ...] the reciprocals of the positive integers, A000027.
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.4, p. 23.
LINKS
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 562.
Wikipedia, Continued Fraction Expansions of Pi.
FORMULA
Equals 2 / (Pi - 2).
Equals 1/arctan(cot(1)). - Daniel Hoyt, Apr 11 2020
From Stefano Spezia, Oct 26 2024: (Start)
2/(Pi - 2) = 1 + K_{n>=1} n*(n+1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 2/(Pi - 2) = 1 + 1*2/(1 + 2*3/(1 + 3*4/(1 + 4*5/(1 + 5*6/(1 + ...))))) (see Finch at p. 23).
2/(Pi - 2) = Sum_{n>=1} (2/Pi)^n (see Shamos). (End)
Equals A309091/2. - Hugo Pfoertner, Oct 28 2024
EXAMPLE
1.7519383938841086612039097015114538792503980068057415636404709501399828870437...
MATHEMATICA
First[RealDigits[2/(Pi - 2), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)
PROG
(PARI) 2 / (Pi - 2) \\ Michel Marcus, Dec 05 2019
(PARI) 1/atan(cotan(1)) \\ Daniel Hoyt, Apr 11 2020
CROSSREFS
KEYWORD
AUTHOR
Daniel Hoyt, Dec 03 2019
STATUS
approved