A196608 Decimal expansion of the least x>0 satisfying 1=x*cos(2*x).

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

Offset

1,1

Comments

Any solution other than 1 or 0 to an equation of the form x=t(f(x)) where t is a trigonometric function and f returns algebraic values for algebraic arguments is transcendental by the Lindemann-Weierstrass theorem. This means that all the solutions to the above equation as well as those in A196602, A196609 and A196626 are transcendental. - Chayim Lowen, Aug 15 2015

Links

Table of n, a(n) for n=1..100.

Index entries for transcendental numbers

Formula

x is the limit of the iteration of x -> Pi - arccos(1/x)/2 on an initial argument a such that abs(a)>=1. - Chayim Lowen, Aug 16 2015

Example

x=2.55709109392790793745988777446340038675281809990...

Mathematica

Plot[{1/x, Cos[x], Cos[2 x], Cos[3 x], Cos[4 x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[1/x == Cos[x], {x, 4, 7}, WorkingPrecision -> 100]
RealDigits[t]  (* A133868 *)
t = x /. FindRoot[1/x == Cos[2 x], {x, 2, 3}, WorkingPrecision -> 100]
RealDigits[t]  (* A196608 *)
t = x /. FindRoot[1/x == Cos[3 x], {x, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196602 *)
t = x /. FindRoot[1/x == Cos[4 x], {x, .9, 1.4}, WorkingPrecision -> 100]
RealDigits[t]  (* A196609 *)
t = x /. FindRoot[1/x == Cos[5 x], {x, .9, 1.2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196626 *)

Crossrefs

Cf. A196602, A196609, A196626.

Sequence in context: A117899 A120839 A332525 * A129228 A228587 A021395

Adjacent sequences: A196605 A196606 A196607 * A196609 A196610 A196611

Keyword

nonn,cons

Author

Clark Kimberling, Oct 05 2011

Status

approved