A249022 -id:A249022 - cp's OEIS frontend

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

Clark Kimberling, Oct 22 2014

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.

			Tangent Euler constant = 0.9943228747334639280815980...

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

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 *)

A362752 Decimal expansion of Sum_{k>=1} (1/k - sin(1/k)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 02 2023

			0.19189908550626482798114607722643984340430910237755...

Math Stackexchange, Closed Form for Sum_{k=1..n} sin(1/k), 2011.

Cf. A233383, A248945, A248946, A249022, A362753.

evalf(sum(1/k - sin(1/k), k = 1..infinity), 120);

sumalt(k = 1, (-1)^(k+1) * zeta(2*k+1)/(2*k+1)!)

Equals Sum_{k>=1} (-1)^(k+1)*zeta(2*k+1)/(2*k+1)!.

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A249023 Decimal expansion of Tangent Euler constant.

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A362752 Decimal expansion of Sum_{k>=1} (1/k - sin(1/k)).

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