A327773 Decimal expansion of Sum_{k>=1} 1/(k*(k+1))^4.

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

Offset: 0

Author

Vaclav Kotesovec, Sep 25 2019

Keywords

• nonn
• cons

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)).

Examples

			0.06332780438680511248031072600283958992849992797342257007711701828839...
		

Links

• R. J. Mathar, Tighly circumscribed regular polygons, arXiv:1301.6293 [math.MG], 2013, see Appendix.
• Eric Weisstein's World of Mathematics, Bernoulli Number
• Eric Weisstein's World of Mathematics, Riemann zeta function

Crossrefs

Cf. A145426, A248619.

Programs

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]]

PARI

suminf(k=1, 1/(k*(k+1))^4)

Formula

Equals Pi^4/45 + 10*Pi^2/3 - 35.