A196500 Decimal expansion of the greatest x satisfying x=1/x+cot(1/x).

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, 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, 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

Offset: 0

Author

Clark Kimberling, Oct 03 2011

Keywords

• nonn
• cons

Comments

Let B be the greatest x satisfying x=1/x+cot(1/x), so that B=0.364... Then

...

cot(1/x) < x < 1/x+cot(1/x) for all x > B; equivalently,

...

cot(x) < 1/x < x+cot(x) for 0 < x < 1/B = 2.7437....

...

These inequalities and those at A196503 supplement the trigonometric inequalities given in Bullen's dictionary cited below.

Examples

			B=0.364470368591040538004400214637816084912410...
1/B=2.7437072699922693825611220811203071372042...
		

References

• P. S. Bullen, A Dictionary of Inequalities, Longman, 1998, pages 250-251.

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A196501, A196502, A196504.

Programs

Mathematica

Plot[{Cot[1/x], x, 1/x + Cot[1/x]}, {x, 0.34, 1.0}]
t = x /.FindRoot[1/x + Cot[1/x] == x, {x, .3, .4}, WorkingPrecision -> 100]
RealDigits[t] (* A196500 *)
1/t
RealDigits[%] (* A196501 *)