A269574 - OEIS

%I #25 Mar 27 2024 20:11:17

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

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

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

%N Decimal expansion of Sum_{n>=1} (1-cos(Pi/n)).

%C Value very close to A193081.

%F Equals 2 * Sum_{n>=1} (sin(Pi/(2*n)))^2.

%F Equals Sum_{k>=1} (-1)^(k+1) * Pi^(2*k) * Zeta(2*k) / (2*k)!, where Zeta is the Riemann zeta function.

%F Equals Sum_{k>=1} 2^(2*k-1) * Pi^(4*k) * B(2*k) / (2*k)!^2, where B(n) is the Bernoulli number A027641(n)/A027642(n).

%e 4.87071896189479740325580288922801180768723798317416757630477557161789...

%p evalf(Sum(1-cos(Pi/n), n=1..infinity), 120);

%t RealDigits[NSum[1 - Cos[Pi/n], {n, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]] (* Be aware that NSum[1 - Cos[Pi/n], {n, 1, Infinity}, WorkingPrecision -> 120] or N[Sum[1 - Cos[Pi/n], {n, 1, Infinity}], 120] give an incorrect numerical result (only 25 decimal places are correct!) *)

%o (PARI) default(realprecision,120); sumpos(n=1, 1-cos(Pi/n))

%Y Cf. A051762, A085365, A093721, A193081, A269611, A269720.

%K nonn,cons

%O 1,1

%A _Vaclav Kotesovec_, Mar 01 2016