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    Bailey-Borwein-Plouffe Formula

    Bailey-Borwein-Plouffe formula (common versions)

    log(3) = sum_(k=0)^∞ 1/((2 k + 1) 4^k) log(2) = 1/3 ( sum_(k=0)^∞ 1/(9^k (1 + 2 k))) 2 log(3) = 1/9 sum_(k=0)^∞ (9/(1 + 2 k) + 1/(2 + 2 k))/9^k log((z^2 + z + 1)/(z^2 - 2 z + 1)) = (( sum_(k=0)^∞ (z/(1 + 3 k) + 1/(2 + 3 k))/z^(3 k)) 3)/z^2 for abs(z)>1 tan^(-1)(2) = 1/8 sum_(k=0)^∞ (8/(1 + 8 k) + 4/(3 + 8 k) - 2/(5 + 8 k) - 1/(7 + 8 k))/16^k log(3) = 1/729 sum_(k=0)^∞ (729/(1 + 6 k) + 81/(2 + 6 k) + 81/(3 + 6 k) + 9/(4 + 6 k) + 9/(5 + 6 k) + 1/(6 + 6 k))/729^k log^2(2) = 1/3 sum_(k=0)^∞(2^(3 - 4 k)/(1 + 8 k)^2 - (5 2^(2 - 4 k))/(2 + 8 k)^2 - 2^(2 - 4 k)/(3 + 8 k)^2 - (7 2^(1 - 4 k))/(4 + 8 k)^2 - 2^(1 - 4 k)/(5 + 8 k)^2 - 5/(2^(4 k) (6 + 8 k)^2) + 1/(2^(4 k) (7 + 8 k)^2) - (3 2^(-1 - 4 k))/(8 + 8 k)^2) log^3(2) = 3 sum_(k=0)^∞(2^(5 - 4 k)/(2 + 8 k)^3 + 2^(6 - 4 k)/(4 + 8 k)^3 + 2^(3 - 4 k)/(6 + 8 k)^3 + 2^(6 - 4 k)/(8 + 8 k)^3) - sum_(k=0)^∞(2^(4 - 12 k)/(1 + 8 k)^3 - (3 2^(1 - 12 k))/(2 + 8 k)^3 - 2^(1 - 12 k)/(3 + 8 k)^3 - 2^(-2 - 12 k)/(4 + 8 k)^3 - 2^(-2 - 12 k)/(5 + 8 k)^3 - (3 2^(-5 - 12 k))/(6 + 8 k)^3 + 2^(-5 - 12 k)/(7 + 8 k)^3 + (7 2^(-8 - 12 k))/(8 + 8 k)^3) ζ(3) = 1/7 ( sum_(k=0)^∞(2^(3 - 4 k)/(1 + 8 k)^3 - (7 2^(3 - 4 k))/(2 + 8 k)^3 - 2^(2 - 4 k)/(3 + 8 k)^3 + (5 2^(3 - 4 k))/(4 + 8 k)^3 - 2^(1 - 4 k)/(5 + 8 k)^3 - (7 2^(1 - 4 k))/(6 + 8 k)^3 + 1/(2^(4 k) (7 + 8 k)^3))) 3 + 1/7 sum_(k=0)^∞(2^(2 - 12 k)/(1 + 8 k)^3 + 2^(1 - 12 k)/(2 + 8 k)^3 - 2^(-1 - 12 k)/(3 + 8 k)^3 - 2^(-1 - 12 k)/(4 + 8 k)^3 - 2^(-4 - 12 k)/(5 + 8 k)^3 + 2^(-5 - 12 k)/(6 + 8 k)^3 + 2^(-7 - 12 k)/(7 + 8 k)^3) Catalan = 3 sum_(k=0)^∞ (1/(2 (8 k + 1)^2) - 1/(2 (8 k + 2)^2) + 1/(2^2 (8 k + 3)^2) - 1/(2^3 (8 k + 5)^2) + 1/(2^3 (8 k + 6)^2) - 1/(2^4 (8 k + 7)^2))/2^(4 k) - 2 sum_(k=0)^∞ (1/(2^3 (8 k + 1)^2) + 1/(2^4 (8 k + 2)^2) + 1/(2^6 (8 k + 3)^2) - 1/(2^9 (8 k + 5)^2) - 1/(2^10 (8 k + 6)^2) - 1/(2^12 (8 k + 7)^2))/2^(12 k) log^4(2) = 1/615 ( sum_(k=0)^∞((73 2^(7 - 4 k))/(1 + 8 k)^4 - (2617 2^(7 - 4 k))/(2 + 8 k)^4 - (73 2^(6 - 4 k))/(3 + 8 k)^4 - (2533 2^(7 - 4 k))/(4 + 8 k)^4 - (73 2^(5 - 4 k))/(5 + 8 k)^4 - (2617 2^(5 - 4 k))/(6 + 8 k)^4 + (73 2^(4 - 4 k))/(7 + 8 k)^4 - (6891 2^(6 - 4 k))/(8 + 8 k)^4)) 3 + 1/615 sum_(k=0)^∞((629 2^(6 - 12 k))/(1 + 8 k)^4 - (761 2^(4 - 12 k))/(2 + 8 k)^4 - (629 2^(3 - 12 k))/(3 + 8 k)^4 - (497 2^(1 - 12 k))/(4 + 8 k)^4 - 629/(2^(12 k) (5 + 8 k)^4) - (761 2^(-2 - 12 k))/(6 + 8 k)^4 + (629 2^(-3 - 12 k))/(7 + 8 k)^4 + (2019 2^(-5 - 12 k))/(8 + 8 k)^4) ζ(5) = (( sum_(k=0)^∞((31 2^(10 - 4 k))/(1 + 8 k)^5 - (807 2^(11 - 4 k))/(2 + 8 k)^5 - (31 2^(9 - 4 k))/(3 + 8 k)^5 - (1553 2^(11 - 4 k))/(4 + 8 k)^5 - (31 2^(8 - 4 k))/(5 + 8 k)^5 - (807 2^(9 - 4 k))/(6 + 8 k)^5 + (31 2^(7 - 4 k))/(7 + 8 k)^5 + (9319 2^(10 - 4 k))/(8 + 8 k)^5)) 9)/62651 + (( sum_(k=0)^∞((173 2^(8 - 12 k))/(1 + 8 k)^5 + (71 2^(9 - 12 k))/(2 + 8 k)^5 - (173 2^(5 - 12 k))/(3 + 8 k)^5 - (457 2^(4 - 12 k))/(4 + 8 k)^5 - (173 2^(2 - 12 k))/(5 + 8 k)^5 + (71 2^(3 - 12 k))/(6 + 8 k)^5 + (173 2^(-1 - 12 k))/(7 + 8 k)^5 - (111 2^(-2 - 12 k))/(8 + 8 k)^5)) 7)/62651 - (( sum_(k=0)^∞(2^(6 - 20 k)/(1 + 8 k)^5 - 2^(1 - 20 k)/(3 + 8 k)^5 - 2^(-1 - 20 k)/(4 + 8 k)^5 - 2^(-4 - 20 k)/(5 + 8 k)^5 + 2^(-20 - 20 k)/(7 + 8 k)^5 + 2^(-11 - 20 k)/(8 + 8 k)^5)) 738)/62651 log^5(2) = ( sum_(k=0)^∞((2783 2^(7 - 4 k))/(1 + 8 k)^5 - (32699 2^(10 - 4 k))/(2 + 8 k)^5 - (2783 2^(6 - 4 k))/(3 + 8 k)^5 - (187547 2^(9 - 4 k))/(4 + 8 k)^5 - (2783 2^(5 - 4 k))/(5 + 8 k)^5 - (32699 2^(8 - 4 k))/(6 + 8 k)^5 + (2783 2^(4 - 4 k))/(7 + 8 k)^5 + (52183 2^(13 - 4 k))/(8 + 8 k)^5))/2021 + ( sum_(k=0)^∞((29537 2^(5 - 12 k))/(1 + 8 k)^5 + (39723 2^(5 - 12 k))/(2 + 8 k)^5 - (29537 2^(2 - 12 k))/(3 + 8 k)^5 - (108983 2^(1 - 12 k))/(4 + 8 k)^5 - (29537 2^(-1 - 12 k))/(5 + 8 k)^5 + (39723 2^(-1 - 12 k))/(6 + 8 k)^5 + (29537 2^(-4 - 12 k))/(7 + 8 k)^5 - (49909 2^(-5 - 12 k))/(8 + 8 k)^5))/2021 - (( sum_(k=0)^∞(2^(4 - 20 k)/(1 + 8 k)^5 - 2^(-1 - 20 k)/(3 + 8 k)^5 - 2^(-3 - 20 k)/(4 + 8 k)^5 - 2^(-6 - 20 k)/(5 + 8 k)^5 + 2^(-11 - 20 k)/(7 + 8 k)^5 + 2^(-13 - 20 k)/(8 + 8 k)^5)) 13199)/2021

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