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%I #10 Mar 27 2024 20:11:09
%S 4,0,9,6,4,3,4,8,9,1,5,0,1,7,3,9,8,3,2,2,2,0,2,3,4,5,8,8,6,2,6,0,5,5,
%T 4,9,5,9,2,8,1,4,4,1,6,5,1,1,9,1,2,0,4,7,5,6,4,4,4,8,6,6,4,0,6,3,9,7,
%U 5,1,0,4,3,5,0,6,7,8,0,7,8,1,7,0,5,8,2,2,8,1,6,9,6,0,8,5,0,4,4,9,8,7,4,7,0
%N Decimal expansion of Sum_{n>=1} (Pi/n - sin(Pi/n)).
%F Equals Sum_{k>=2} (-1)^k * Pi^(2*k-1) * Zeta(2*k-1) / (2*k-1)!, where Zeta is the Riemann zeta function.
%e 4.096434891501739832220234588626055495928144165119120475644486640639751...
%p evalf(Sum(Pi/n - sin(Pi/n), n=1..infinity), 120);
%t RealDigits[NSum[Pi/n - Sin[Pi/n], {n, 1, Infinity}, WorkingPrecision->200, NSumTerms->10000, PrecisionGoal->120, Method->{"NIntegrate", "MaxRecursion"->100}]][[1]]
%t (* Be aware that N[Sum[Pi/n - Sin[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, Pi/n - sin(Pi/n))
%Y Cf. A269574, A269611, A093720, A269768.
%K nonn,cons
%O 1,1
%A _Vaclav Kotesovec_, Mar 04 2016
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