A100554 Decimal expansion of the fractional part of Sum_{n>=1} cos((n + 1)*Pi)*zeta(2*n) = zeta(2) - zeta(4) + zeta(6) - zeta(8) + ..., where Zeta is the Riemann zeta function.

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

Offset: 0

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 27 2004

Keywords

• cons
• nonn

Comments

For odd upper bounds, the sum converges to the given value p in (0,1) with no fractional part function necessary. For even upper bounds, the sum converges to p+1.

Decimal expansion of (psi(i)-psi(-i))/2/i-3/2 where psi is the digamma function. - Benoit Cloitre, Nov 28 2004

Examples

			0.576674047468581174134050794750000490...
		

Crossrefs

Cf. A000796.

Programs

Maple

evalf(Pi*coth(Pi)/2-1) ; # R. J. Mathar, Apr 01 2010

Mathematica

N[FractionalPart[Sum[Cos[(n + 1)*Pi]*Zeta[2*n], {n, 1000}]], 140]
RealDigits[Pi*Coth[Pi]/2 - 1, 10, 105] // First (* Jean-François Alcover, Jan 06 2014, after R. J. Mathar *)

PARI

(psi(I)-psi(-I))/2/I-3/2

PARI

sumnumrat(1/(x^2+1),2) \\ Charles R Greathouse IV, Jan 20 2022

PARI

sumnumrat(1/(x^2+4*x+5),0) \\ Charles R Greathouse IV, Jan 20 2022

Formula

Equals Pi*(coth(Pi))/2 -1 where Pi = A000796. - R. J. Mathar, Apr 01 2010

Equals Sum_{k>=2} 1/(k^2 + 1). - Amiram Eldar, Aug 15 2020