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