OFFSET
0,1
COMMENTS
In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives f(4).
FORMULA
Equals (-1 + (2*Pi)*coth(2*Pi))/8 = (-1 + (2*Pi*i)*cot(2*Pi*i))/8, i = sqrt(-1).
Equals Integral_{x=0..oo} sin(x)*cos(x)/(exp(x) - 1) dx. - Amiram Eldar, Aug 16 2020
EXAMPLE
Sum_{k>=1} 1/(k^2+4) = 0.66040364132111511419...
MATHEMATICA
RealDigits[(-1 + 2*Pi*Coth[2*Pi])/8, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)
PROG
(PARI) default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(4)
(PARI) sumnumrat(1/(x^2+4), 1) \\ Charles R Greathouse IV, Jan 20 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Nov 04 2019
STATUS
approved