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Decimal expansion of the second smallest positive root of tan(x) = x.
1

%I #22 Jul 11 2024 09:40:03

%S 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,

%T 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,

%U 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

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

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

%C 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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/duBois-ReymondConstants.html">du Bois-Reymond Constants</a>.

%e 7.72525183693770716419506893306298662637815930461...

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

%o (PARI) solve(x=7,7.8,tan(x)-x) \\ _Charles R Greathouse IV_, Apr 20 2016

%Y 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).

%Y Cf. A079330, A088989.

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Feb 20 2015