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Decimal expansion of Sum_{k>=0} arctan(1/2^k).
1

%I #28 Nov 15 2024 23:36:13

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

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

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

%N Decimal expansion of Sum_{k>=0} arctan(1/2^k).

%C This number can be interpreted geometrically as the angle in radians of a fan made of stacked right triangles, with the length to height ratio doubling each successive triangle as seen in the illustration.

%C Since this angle exceeds Pi/2, the set of rotation angles used in the CORDIC algorithm covers an angle range sufficient to compute sine and cosine for any angle between 0 and Pi/2. This means the algorithm can converge to any angle in that range through appropriate combinations of these basic rotations. - _Daniel Hoyt_, Oct 25 2024

%H Daniel Hoyt, <a href="/A344906/a344906.png">Illustration of this angle's arctan relationship</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/CORDIC#Rotation_mode">CORDIC</a>.

%F Equals Sum_{k>=1} (-1)^(k+1)*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)).

%e 1.743286620472340003...

%p Digits:= 140:

%p evalf(sum(arccot(2^k), k=0..infinity)); # _Alois P. Heinz_, Jun 02 2021

%o (PARI) suminf(k=0, atan(1/2^k))

%o (PARI) sumalt(k=1, ((-1)^(k+1))*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)))

%Y Cf. A003881, A065445, A073000, A195727, A195782.

%K nonn,cons

%O 1,2

%A _Daniel Hoyt_, Jun 01 2021