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 A327773 Decimal expansion of Sum_{k>=1} 1/(k*(k+1))^4. 1
 0, 6, 3, 3, 2, 7, 8, 0, 4, 3, 8, 6, 8, 0, 5, 1, 1, 2, 4, 8, 0, 3, 1, 0, 7, 2, 6, 0, 0, 2, 8, 3, 9, 5, 8, 9, 9, 2, 8, 4, 9, 9, 9, 2, 7, 9, 7, 3, 4, 2, 2, 5, 7, 0, 0, 7, 7, 1, 1, 7, 0, 1, 8, 2, 8, 8, 3, 9, 0, 6, 4, 0, 4, 3, 7, 9, 5, 5, 1, 6, 9, 7, 8, 6, 3, 9, 7, 2, 8, 4, 2, 7, 8, 5, 7, 3, 9, 4, 0, 5, 4, 6, 0, 6, 8, 0 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Sum_{k>=1} 1/(k*(k+1)) = 1 Sum_{k>=1} 1/(k*(k+1))^2 = -3 + Pi^2/3 Sum_{k>=1} 1/(k*(k+1))^3 = 10 - Pi^2 Sum_{k>=1} 1/(k*(k+1))^4 = -35 + 10*Pi^2/3 + Pi^4/45 Sum_{k>=1} 1/(k*(k+1))^5 = 126 - 35*Pi^2/3 - Pi^4/9 Sum_{k>=1} 1/(k*(k+1))^6 = -462 + 42*Pi^2 + 7*Pi^4/15 + 2*Pi^6/945 Sum_{k>=1} 1/(k*(k+1))^7 = 1716 - 154*Pi^2 - 28*Pi^4/15 - 2*Pi^6/135 Sum_{k>=1} 1/(k*(k+1))^8 = -6435 + 572*Pi^2 + 22*Pi^4/3 + 8*Pi^6/105 + Pi^8/4725 Sum_{k>=1} 1/(k*(k+1))^9 = 24310 - 2145*Pi^2 - 143*Pi^4/5 - 22*Pi^6/63 - Pi^8/525 Sum_{k>=1} 1/(k*(k+1))^10 = -92378 + 24310*Pi^2/3 + 1001*Pi^4/9 + 286*Pi^6/189 + 11*Pi^8/945 + 2*Pi^10/93555 In general, for s > 1, Sum_{k>=1} 1/(k*(k+1))^s = (-1)^s * (-binomial(2*s-1, s-1) + Sum_{j=1..floor(s/2)} ((-1)^(j+1) * binomial(2*s-2*j-1, s-1) * Bernoulli(2*j) * (2*Pi)^(2*j) / (2*j)!). Equivalently, for s > 1, Sum_{k>=1} 1/(k*(k+1))^s = (-1)^s * (-binomial(2*s-1, s-1) + 2*Sum_{j=1..floor(s/2)} (binomial(2*s-2*j-1, s-1) * zeta(2*j)). LINKS R. J. Mathar, Tighly circumscribed regular polygons, arXiv:1301.6293 (2013), see Appendix. Eric Weisstein's World of Mathematics, Bernoulli Number Eric Weisstein's World of Mathematics, Riemann zeta function FORMULA Equals Pi^4/45 + 10*Pi^2/3 - 35. EXAMPLE 0.06332780438680511248031072600283958992849992797342257007711701828839... MAPLE evalf(sum(1/(k*(k+1))^4, k=1..infinity), 120); MATHEMATICA RealDigits[N[Sum[1/(k*(k + 1))^4, {k, 1, Infinity}], 105]][[1]] PROG (PARI) suminf(k=1, 1/(k*(k+1))^4) CROSSREFS Cf. A145426, A248619. Sequence in context: A144542 A222457 A198872 * A085670 A011410 A298206 Adjacent sequences:  A327770 A327771 A327772 * A327774 A327775 A327776 KEYWORD nonn,cons AUTHOR Vaclav Kotesovec, Sep 25 2019 STATUS approved

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Last modified June 14 15:19 EDT 2021. Contains 345025 sequences. (Running on oeis4.)