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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.
4
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
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
KEYWORD
cons,nonn
AUTHOR
Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 27 2004
STATUS
approved