A196608 - OEIS

%I #19 May 14 2019 21:26:54

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

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

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

%N Decimal expansion of the least x>0 satisfying 1=x*cos(2*x).

%C Any solution other than 1 or 0 to an equation of the form x=t(f(x)) where t is a trigonometric function and f returns algebraic values for algebraic arguments is transcendental by the Lindemann-Weierstrass theorem. This means that all the solutions to the above equation as well as those in A196602, A196609 and A196626 are transcendental. - _Chayim Lowen_, Aug 15 2015

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

%F x is the limit of the iteration of x -> Pi - arccos(1/x)/2 on an initial argument a such that abs(a)>=1. - _Chayim Lowen_, Aug 16 2015

%e x=2.55709109392790793745988777446340038675281809990...

%t Plot[{1/x, Cos[x], Cos[2 x], Cos[3 x], Cos[4 x]}, {x, 0, 2 Pi}]

%t t = x /. FindRoot[1/x == Cos[x], {x, 4, 7}, WorkingPrecision -> 100]

%t RealDigits[t]  (* A133868 *)

%t t = x /. FindRoot[1/x == Cos[2 x], {x, 2, 3}, WorkingPrecision -> 100]

%t RealDigits[t]  (* A196608 *)

%t t = x /. FindRoot[1/x == Cos[3 x], {x, 1, 2}, WorkingPrecision -> 100]

%t RealDigits[t]  (* A196602 *)

%t t = x /. FindRoot[1/x == Cos[4 x], {x, .9, 1.4}, WorkingPrecision -> 100]

%t RealDigits[t]  (* A196609 *)

%t t = x /. FindRoot[1/x == Cos[5 x], {x, .9, 1.2}, WorkingPrecision -> 100]

%t RealDigits[t]  (* A196626 *)

%Y Cf. A196602, A196609, A196626.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 05 2011