A249023 Decimal expansion of Tangent Euler constant.

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

Offset: 0

Author

Clark Kimberling, Oct 22 2014

Keywords

• nonn
• cons
• more

Comments

The Tangent-Euler constant is introduced here as the limit as n increases without bound of Sum_{k=1..n} tan(1/k) - Integral_{x=1..n} tan(1/x) dx; this is analogous to the Euler constant, defined as the limit of Sum_{k=1..n} 1/k - Integral_{x=1..n} 1/x dx.

Examples

			Tangent Euler constant = 0.9943228747334639280815980...
		

Crossrefs

Cf. A001620 (Euler constant), A249022 (Sine Euler constant).

Programs

Mathematica

CI = N[Integrate[Normal[Series[Tan[x], {x, 0, 500}]] /. x -> 1/x, x] /. x -> 1, 100]; t = (Total[Table[((-1)^(n - 1) 2^(2 n) (2^(2 n) - 1) BernoulliB[2 n])/(2 n)! HarmonicNumber[k, 2 n - 1], {n, 80}]] /. k -> #) - (N[Integrate[Normal[Series[Tan[x], {x, 0, 10}]] /. x -> 1/x, x] /. x -> #, 100] - CI) &[N[10^30, 30]]
RealDigits[t][[1]]
(* Peter J. C. Moses, Oct 20 2014 *)