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    Euler Sum

    Euler's sum (common versions)

    sum_(k=1)^∞ (H_k)^2/(k + 1)^2 = (11 π^4)/360 sum_(k=1)^∞ (H_k)^2/k^2 = (17 π^4)/360 sum_(k=1)^∞ (H_k)^2/(k + 1)^4 = (37 π^6)/22680 - ζ(3)^2 sum_(k=1)^∞ (H_k)^4/(1 + k)^2 = (859 π^6)/22680 + 3 ζ(3)^2 sum_(k=1)^∞ (H_k)^2/k^4 = (97 π^6)/22680 - 2 ζ(3)^2 sum_(k=1)^∞ (H_k)^3/(1 + k)^3 = -(11 π^6)/5040 + 2 ζ(3)^2 sum_(k=1)^∞ (( sum_(j=1)^k (-1)^(j + 1)/j^2) (-1)^(k + 1))/(k + 1)^4 = -(97 π^6)/90720 + (3 ζ(3)^2)/4 sum_(k=1)^∞ (H_k)^3/(1 + k)^2 = 1/6 (π^2 ζ(3) + 45 ζ(5)) sum_(k=1)^∞ (H_k)^2/(k + 1)^3 = 1/6 (π^2 ζ(3) - 9 ζ(5)) sum_(k=1)^∞ (H_k)^2/k^3 = 1/6 (π^2 ζ(3)) (-1) + (7 ζ(5))/2 sum_(k=1)^∞ (( sum_(j=1)^k (-1)^(j + 1)/j) (-1)^(k + 1))/(k + 1)^3 = 1/180 (-π^4 - 15 π^2 log^2(2) + 15 log^4(2)) + 2 Li_4(1/2) sum_(k=1)^∞ ( sum_(j=1)^k (-1)^(j + 1)/j)^2/(k + 1)^2 = -1/720 (29 π^4) + 1/4 π^2 log^2(2) + (log^4(2))/4 + 6 Li_4(1/2) sum_(k=1)^∞ (H_k)^3/(1 + k)^4 = 1/240 (-22 π^4 ζ(3) + 80 π^2 ζ(5) + 1785 ζ(7)) sum_(k=1)^∞ (H_k)^2/k^5 = 1/36 (-π^4 ζ(3) - 6 π^2 ζ(5) + 216 ζ(7)) sum_(k=1)^∞ (H_k)^2/(k + 1)^5 = 1/180 (π^4 ζ(3)) (-1) + (π^2 ζ(5))/6 - ζ(7) sum_(k=1)^∞ (H_k)^5/(1 + k)^2 = 1/30 (π^4 ζ(3)) 11 + 1/4 (π^2 ζ(5)) 19 + (1855 ζ(7))/16 sum_(k=1)^∞ ((-1)^(k + 1) (H_k)^2)/(k + 1)^2 = 1/480 (11 π^4 + 40 π^2 log^2(2) - 40 (log^4(2) + 24 Li_4(1/2) + 21 log(2) ζ(3))) sum_(k=1)^∞ (H_k)^4/(1 + k)^3 = 1/180 (π^4 ζ(3)) 37 - (5 π^2 ζ(5))/6 - (109 ζ(7))/8 sum_(k=1)^∞ (( sum_(j=1)^k (-1)^(j + 1)/j)^2 (-1)^(k + 1))/(k + 1)^2 = (37 π^4)/1440 - 1/3 π^2 log^2(2) - (log^4(2))/6 - 4 Li_4(1/2) + 1/4 (log(2) ζ(3)) 7 sum_(k=1)^∞ H_k/(k + 1)^n = (ζ(n + 1) n)/2 - 1/2 sum_(k=1)^(n - 2) ζ(n - k) ζ(k + 1) for (n element Z and n>1) sum_(k=1)^∞ H_k/k^n = 1/2 ζ(n + 1) (n + 2) - 1/2 sum_(k=1)^(n - 2) ζ(n - k) ζ(k + 1) for (n element Z and n>1) sum_(k=1)^∞ (( sum_(j=1)^k (-1)^(j + 1)/j)^2 (-1)^(k + 1))/(k + 1)^3 = 1/720 (π^4 log(2)) 11 + (log^5(2))/6 + log(16) Li_4(1/2) + 1/48 π^2 (-8 log^3(2) + 3 ζ(3)) - (79 ζ(5))/32 sum_(k=1)^∞ (H_k)^3/(1 + k)^6 = -(37 π^6 ζ(3))/7560 + ζ(3)^3 - (11 π^4 ζ(5))/120 + (π^2 ζ(7))/2 + (197 ζ(9))/24 sum_(k=1)^∞ ( sum_(j=1)^k (-1)^(j + 1)/j)^2/(k + 1)^3 = 4 Li_5(1/2) + 1/720 (-11 π^4 log(2) + 40 π^2 log^3(2) - 24 log^5(2) - 90 (π^2 - 14 log^2(2)) ζ(3)) - (17 ζ(5))/32 sum_(k=1)^∞ (H_k)^5/(1 + k)^4 = -(859 π^6 ζ(3))/4536 - 5 ζ(3)^3 + 1/15 (π^4 ζ(5)) 11 + 1/48 (π^2 ζ(7)) 265 + (890 ζ(9))/9 sum_(k=1)^∞ (H_k)^4/(1 + k)^5 = (11 (π^6 ζ(3)))/1260 - (8 ζ(3)^3)/3 + 1/180 (π^4 ζ(5)) 37 - (7 π^2 ζ(7))/6 - (29 ζ(9))/2 sum_(k=1)^∞ (H_k)^6/(1 + k)^3 = 1/420 (π^6 ζ(3)) 233 + (67 ζ(3)^3)/3 - (27 π^4 ζ(5))/10 - (217 π^2 ζ(7))/16 - (3073 ζ(9))/12 sum_(k=1)^∞ ((-1)^(k + 1) (H_k)^2)/(k + 1)^3 = 1/9 π^2 log^3(2) - (2 log^5(2))/15 - 4 log(2) Li_4(1/2) - 4 Li_5(1/2) + 1/16 (π^2 - 28 log^2(2)) ζ(3) + (107 ζ(5))/32 sum_(k=1)^∞ ( sum_(j=1)^k (-1)^(j + 1)/j)^3/(k + 1)^2 = 1/96 (π^4 log(2)) (-1) 19 + (9 log^5(2))/20 + log(64) Li_4(1/2) - 24 Li_5(1/2) + 1/12 π^2 (5 log^3(2) + ζ(3)) + (659 ζ(5))/32 sum_(k=1)^∞ (H_k)^7/(1 + k)^2 = (134701 ζ(9))/36 + 1/8 (ζ(4) ζ(5)) 15697 + 1/24 (ζ(3) ζ(6)) 29555 + 56 ζ(3)^3 + 1/4 (ζ(2) ζ(7)) 3287 sum_(k=1)^∞ (H_k)^2/(k + 1)^(2 n + 1) = 1/3 ζ(2 n + 1) π^2 + 1/6 ζ(2 n + 3) (-8 - 3 n + 2 n^2) - 1/2 ( sum_(k=0)^(2 n - 1) ζ(2 n - k + 1) ζ(k + 2)) (2 n + 1) + 1/3 sum_(k=2)^(2 n - 1) ζ(2 n - k + 1) sum_(j=1)^(k - 1) ζ(j + 1) ζ(k - j + 1) + 2 sum_(k=1)^n k ζ(2 k + 1) ζ(2 n - 2 k + 2) for (n element Z and n>0) sum_(k=1)^∞ (H_k)^2/k^(2 n + 1) = 1/6 (10 + 9 n + 2 n^2) ζ(2 n + 3) + 1/3 ζ(2 n + 1) π^2 - (n + 3/2) sum_(k=1)^(2 n) ζ(2 n + 2 - k) ζ(k + 1) + 1/3 sum_(k=2)^(2 n - 1) ζ(2 n - k + 1) sum_(j=1)^(k - 1) ζ(j + 1) ζ(k - j + 1) + 2 sum_(k=1)^n k ζ(2 k + 1) ζ(2 n - 2 k + 2) for (n element Z and n>0)

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