A196504 - OEIS

%I #24 Feb 03 2025 12:47:43

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

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

%U 6,6,5,0,3,1,0,4,2,7,6,2,3,4,5,9,2,2,7,9,5,1,5,0,4,2,5,6,3,0,6,3,9

%N Decimal expansion of the least x > 0 satisfying x + tan(x) = 0.

%C Let L be the least x > 0 satisfying x + tan(x) = 0.

%C Then L is also the least x > 0 satisfying x = (sin(x))(sqrt(1+x^2)).

%C Consequently, for 0 < x < L, for all p > 0, 1/sqrt(1+x^2) - 1/x^p < sin(x) < 1/sqrt(1+x^2) for 0 < x < L.

%C See A196500-A196503 and A196505 for related constants and inequalities.

%C The number L also occurs in connection with Du Bois Reymond's constants; see the Finch reference.

%C For x = L the area of right triangle with vertices (0,0), (x,0) and (x,sin(x)), i.e., the one inscribed into the half-wave curve, is maximal. - _Roman Witula_, Feb 05 2015

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 239.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e L = 2.02875783811043422357697112473471437610838002...

%e 1/L = 0.4929124517549075741877801898222329769156970132...

%t Plot[{Sin[x], x/Sqrt[1 + x^2]}, {x, 0, 9}]

%t t = x /.FindRoot[Sin[x] == x/Sqrt[1 + x^2], {x, .10, 3}, WorkingPrecision -> 100]

%t RealDigits[t]   (* A196504 *)

%t 1/t

%t RealDigits[1/t] (* A196505 *)

%Y Cf. A196505, A196500, A196502.

%K nonn,cons,changed

%O 1,1

%A _Clark Kimberling_, Oct 03 2011