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Decimal expansion of least x satisfying x^2 + 5 = csc(x) and 0 < x < Pi.
3

%I #12 Jan 30 2025 13:52:09

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

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

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

%N Decimal expansion of least x satisfying x^2 + 5 = 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="/A201570/b201570.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: 0.19974229281947213708674051595534811453...

%e greatest: 3.07227983005125033585986646046469906...

%t a = 1; c = 5;

%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] (* A201570 *)

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

%t RealDigits[r] (* A201571 *)

%o (PARI) a=1; c=5; 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,changed

%O 0,2

%A _Clark Kimberling_, Dec 03 2011