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Decimal expansion of least x satisfying x^2 + 8 = csc(x) and 0 < x < Pi.
3

%I #13 Feb 07 2025 16:44:07

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

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

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

%N Decimal expansion of least x satisfying x^2 + 8 = csc(x) and 0 < x < Pi.

%C See A201564 for a guide to related sequences. The Mathematica program includes a graph.

%H G. C. Greubel, <a href="/A201576/b201576.txt">Table of n, a(n) for n = 0..10000</a>

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

%e least: 0.125081922635997441289177701653785707187...

%e greatest: 3.084464140564380849459190595364646021...

%t a = 1; c = 8;

%t f[x_] := a*x^2 + c; g[x_] := Csc[x]

%t Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, .1, .2}, WorkingPrecision -> 110]

%t RealDigits[r] (* A201576 *)

%t r = x /. FindRoot[f[x] == g[x], {x, 3.0, 3.1}, WorkingPrecision -> 110]

%t RealDigits[r] (* A201577 *)

%o (PARI) a=1; c=8; solve(x=0.1, 0.2, a*x^2 + c - 1/sin(x)) \\ _G. C. Greubel_, Aug 21 2018

%Y Cf. A201564.

%K nonn,cons

%O 0,2

%A _Clark Kimberling_, Dec 03 2011