login
Decimal expansion of arctan(1000000).
11

%I #7 Sep 08 2012 11:30:11

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

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

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

%N Decimal expansion of arctan(1000000).

%C pi/2-arctan(1000000)<1/1000000; the first nonzero digits of pi/2-arctan(1000000) are as follows:

%C 999999999999666666666666866666666666. The twelve 6's before 8 correspond to the limit shown at the end of the Mathematica program. What about the next eleven 6's?

%e Let x=pi/2 and y=arc(1000000); then

%e x=1.57079632679489661923132169163975144209858469968755291048...

%e y=1.57079532679489661956465502497288477543191817587802910085...

%e x-y=0.000000099999999999966666666666686666666666652380963492...

%t N[Pi/2, 100]

%t N[ArcTan[10^6], 100]

%t RealDigits[%] (* A195793 *)

%t Limit[n^2 - (n^3) (Pi/2 - ArcTan[n]), n -> Infinity]

%t (* Limit equals 1/3 *)

%Y Cf. A000796, A195789.

%Y For other approximations to Pi see A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548. - _N. J. A. Sloane_, Sep 08 2012

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Sep 24 2011