A269720 Decimal expansion of Sum_{n>=1} (Pi/n - sin(Pi/n)).

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, 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, 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

Offset: 1

Author

Vaclav Kotesovec, Mar 04 2016

Keywords

• nonn
• cons

Examples

			4.096434891501739832220234588626055495928144165119120475644486640639751...
		

Crossrefs

Cf. A269574, A269611, A093720, A269768.

Programs

Maple

evalf(Sum(Pi/n - sin(Pi/n), n=1..infinity), 120);

Mathematica

RealDigits[NSum[Pi/n - Sin[Pi/n], {n, 1, Infinity}, WorkingPrecision->200, NSumTerms->10000, PrecisionGoal->120, Method->{"NIntegrate", "MaxRecursion"->100}]][[1]]
(* 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! *)

PARI

default(realprecision,120); sumpos(n=1, Pi/n - sin(Pi/n))

Formula

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