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

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, 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, 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

Offset: 1

Author

Vaclav Kotesovec, Mar 01 2016

Keywords

• nonn
• cons

Comments

Value very close to A193081.

Examples

			4.87071896189479740325580288922801180768723798317416757630477557161789...
		

Crossrefs

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

Programs

Maple

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

Mathematica

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!) *)

PARI

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

Formula

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

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

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).