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A269574 Decimal expansion of Sum_{n>=1} (1-cos(Pi/n)).

%I A269574 #25 Mar 27 2024 20:11:17
%S A269574 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 A269574 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 A269574 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 A269574 Decimal expansion of Sum_{n>=1} (1-cos(Pi/n)).
%C A269574 Value very close to A193081.
%F A269574 Equals 2 * Sum_{n>=1} (sin(Pi/(2*n)))^2.
%F A269574 Equals Sum_{k>=1} (-1)^(k+1) * Pi^(2*k) * Zeta(2*k) / (2*k)!, where Zeta is the Riemann zeta function.
%F A269574 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 A269574 4.87071896189479740325580288922801180768723798317416757630477557161789...
%p A269574 evalf(Sum(1-cos(Pi/n), n=1..infinity), 120);
%t A269574 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 A269574 (PARI) default(realprecision,120); sumpos(n=1, 1-cos(Pi/n))
%Y A269574 Cf. A051762, A085365, A093721, A193081, A269611, A269720.
%K A269574 nonn,cons
%O A269574 1,1
%A A269574 _Vaclav Kotesovec_, Mar 01 2016