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

0,1

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

Links

Table of n, a(n) for n=0..104.

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

Example

0.576674047468581174134050794750000490...

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 *)

Prog

(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

Crossrefs

Cf. A000796.

Sequence in context: A114603 A348731 A346590 * A258148 A193013 A220607

Adjacent sequences: A100551 A100552 A100553 * A100555 A100556 A100557

Keyword

cons,nonn

Author

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

Status

approved