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Decimal expansion of the least x>0 satisfying 1/(1+x^2)=3*cos(x).
6

%I #10 Feb 11 2025 13:54:04

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

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

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

%N Decimal expansion of the least x>0 satisfying 1/(1+x^2)=3*cos(x).

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

%e 1.46461147970142500501464804801002599781808481310...

%t Plot[{1/(1 + x^2), Cos[x], 2 Cos[x], 3 Cos[x], 4 Cos[x]}, {x, 0, 2}]

%t t = x /. FindRoot[1 == (1 + x^2) Cos[x], {x, 1, 1.5}, WorkingPrecision -> 100]

%t RealDigits[t] (* A196816 *)

%t t = x /. FindRoot[1 == 2 (1 + x^2) Cos[x], {x, 1, 1.6},

%t WorkingPrecision -> 100]

%t RealDigits[t] (* A196817 *)

%t t = x /. FindRoot[1 == 3 (1 + x^2) Cos[x], {x, 1, 1.6},

%t WorkingPrecision -> 100]

%t RealDigits[t] (* A196818 *)

%t t = x /. FindRoot[1 == 4 (1 + x^2) Cos[x], {x, 1, 1.6},

%t WorkingPrecision -> 100]

%t RealDigits[t] (* A196819 *)

%t t = x /. FindRoot[1 == 5 (1 + x^2) Cos[x], {x, 1, 1.6},

%t WorkingPrecision -> 100]

%t RealDigits[t] (* A196820 *)

%t t = x /. FindRoot[1 == 6 (1 + x^2) Cos[x], {x, 1, 1.6},

%t WorkingPrecision -> 100]

%t RealDigits[t] (* A196821 *)

%Y Cf. A196914.

%K nonn,cons,changed

%O 1,2

%A _Clark Kimberling_, Oct 06 2011