zeta_3_fast_convergence.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 20 May 2017
# https://github.com/trizen

# A very fast converging series for zeta(3).

# The formula is:
#   zeta(3) = -8*pi^2/14 * (log(27/16) + Sum_{n>=0}(zeta(2n) - 1)/((2n + 1) * (2n + 2) * 2^(2n)))

# The formula is based on the following two formulas:
#   -14*zeta(3) / (8*pi^2) = Sum_{n>=0} zeta(2n)/((2n + 1) * (2n + 2) * 2^(2n))
#   log(27/16) = Sum_{n>=0} 1/((2n + 1) * (2n + 2) * 2^(2n))

var sum = log(27/16)

for n in (0..40) {
    sum += (zeta(2*n) - 1)/((2*n + 1) * (2*n + 2) * 2**(2*n))
}

var f = -(8 * Num.pi**2)/14

say sum*f           #=> 1.20205690315959428539973816151144999076498629234
say zeta(3)         #=> 1.20205690315959428539973816151144999076498629234