A330156 Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].

1, 7, 5, 1, 9, 3, 8, 3, 9, 3, 8, 8, 4, 1, 0, 8, 6, 6, 1, 2, 0, 3, 9, 0, 9, 7, 0, 1, 5, 1, 1, 4, 5, 3, 8, 7, 9, 2, 5, 0, 3, 9, 8, 0, 0, 6, 8, 0, 5, 7, 4, 1, 5, 6, 3, 6, 4, 0, 4, 7, 0, 9, 5, 0, 1, 3, 9, 9, 8, 2, 8, 8, 7, 0, 4, 3, 7, 1, 0, 9, 9, 5, 1, 3, 4, 5, 1

Offset: 1

Author

Daniel Hoyt, Dec 03 2019

Keywords

• nonn
• cons

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.

Examples

			1.7519383938841086612039097015114538792503980068057415636404709501399828870437...
		

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.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000027, A000796, A052119, A073824, A309091.

Programs

Mathematica

First[RealDigits[2/(Pi - 2), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

PARI

2 / (Pi - 2) \\ Michel Marcus, Dec 05 2019

PARI

1/atan(cotan(1)) \\ Daniel Hoyt, Apr 11 2020

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