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%I #8 Aug 09 2021 14:00:30
%S 3,6,4,4,7,0,3,6,8,5,9,1,0,4,0,5,3,8,0,0,4,4,0,0,2,1,4,6,3,7,8,1,6,0,
%T 8,4,9,1,2,4,1,0,3,6,4,1,3,0,3,0,2,5,8,1,7,2,1,0,1,5,4,1,0,7,7,8,0,5,
%U 3,6,0,0,5,4,7,1,6,8,2,3,2,2,3,8,5,7,5,3,1,0,4,5,2,4,5,1,7,1,6,2,8,9,9,9
%N Decimal expansion of the greatest x satisfying x=1/x+cot(1/x).
%C Let B be the greatest x satisfying x=1/x+cot(1/x), so that B=0.364... Then
%C ...
%C cot(1/x) < x < 1/x+cot(1/x) for all x > B; equivalently,
%C ...
%C cot(x) < 1/x < x+cot(x) for 0 < x < 1/B = 2.7437....
%C ...
%C These inequalities and those at A196503 supplement the trigonometric inequalities given in Bullen's dictionary cited below.
%D P. S. Bullen, A Dictionary of Inequalities, Longman, 1998, pages 250-251.
%e B=0.364470368591040538004400214637816084912410...
%e 1/B=2.7437072699922693825611220811203071372042...
%t Plot[{Cot[1/x], x, 1/x + Cot[1/x]}, {x, 0.34, 1.0}]
%t t = x /.FindRoot[1/x + Cot[1/x] == x, {x, .3, .4}, WorkingPrecision -> 100]
%t RealDigits[t] (* A196500 *)
%t 1/t
%t RealDigits[%] (* A196501 *)
%Y Cf. A196501, A196502, A196504.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 03 2011
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