A255272 Decimal expansion of the second smallest positive root of tan(x) = x.

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

Offset: 1

Author

Jean-François Alcover, Feb 20 2015

Keywords

• nonn
• cons
• easy

Comments

This constant is quite close to 5*Pi/2 - 1/8 = 7.72898...

Searching for solutions x=k*Pi+Pi/2-e and small e, for k=1,2,3.... means via the approximation tan(x) = 1/e-e/3-e^3/45... that e is approximately 1/(k*Pi+Pi/2), so the constants x are close to k*Pi+Pi/2-1/(k*Pi+Pi/2). Here k=2 and the constant is close to 5*Pi/2-2/(5*Pi) = 7.7266576... - R. J. Mathar, Jul 11 2024

Examples

			7.72525183693770716419506893306298662637815930461...
		

Links

• Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Crossrefs

Cf. A115365 (smallest positive root), A062546 (C_2 = 2nd du Bois-Reymond constant), A224196 (C_3), A207528 (C_4), A243108 (C_5), A245333 (C_6).

Cf. A079330, A088989.

Programs

Mathematica

xi[n_] := x /. FindRoot[Tan[x] == x, {x, n*Pi + Pi/2 - 1/(4*n)}, WorkingPrecision -> 102]; RealDigits[xi[2]] // First

PARI

solve(x=7,7.8,tan(x)-x) \\ Charles R Greathouse IV, Apr 20 2016