%I #8 Oct 27 2014 23:56:08
%S 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
%N Decimal expansion of Tangent Euler constant.
%C The Tangent-Euler constant is introduced here as the limit as n increases without bound of sum{tan(1/k), k = 1..n} - integral{tan(1/x) over [1,n]}; this is analogous to the Euler constant, defined as the limit of sum{1/k, k = 1..n} - integral{1/x over [1,n]}.
%e Tangent Euler constant = 0.9943228747334639280815980...
%t 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]]
%t RealDigits[t][[1]]
%t (* _Peter J. C. Moses_, Oct 20 2014 *)
%Y Cf. A001620 (Euler constant), A249022 (Sine Euler constant).
%K nonn,easy,cons
%O 0,1
%A _Clark Kimberling_, Oct 22 2014