Vallée's Constant

Author

Eric W. Weisstein
August 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/ValleesConstant.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ValleesConstant.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Probabilities

E(L)

(3/4)+(180/(π^4))Sum[(1/((i^2)(j^2))),{i,∞},{j,i+1,2i}]//InputForm

(3/4)+(180/(π^4))NSum[(1/((i^2)(j^2))),{i,∞},{j,i+1,2i}]

(17/4)+(360/(π^4))Sum[((((-1))^i)/((i^2)(j^2))),{i,∞},{j,i}]//InputForm

(17/4)+(360/(π^4))NSum[((((-1))^i)/((i^2)(j^2))),{i,∞},{j,i}]

17-(60/(π^4))(24PolyLog[4,(1/2)]-(Pi^2)((Log[2])^2)+21Zeta[3]Log[2]+((Log[2])^4))//N

(Subscript[p, 1])

Sum[(1/((i^2)(((i+1))^2))),{i,∞}]

(Subscript[p, 2])

Sum[(1/((((i j+1))^2)(((i j+i+1))^2))),{i,∞},{j,∞}]//Timing

-5+(2/3)(π^2)-2Zeta[3]+2Sum[(((-1))^n)(n+1)Zeta[n+4](Zeta[n+2]-1),{n,0,∞}]

-5+(2/3)(π^2)-2Zeta[3]+2NSum[(((-1))^n)(n+1)Zeta[n+4](Zeta[n+2]-1),{n,0,∞}]

(Subscript[p, 3])

Sum[(1/((((i j k+i+k))^2)(((i j k+i j+i+k+1))^2))),{i,∞},{j,∞},{k,∞}]//Timing

NSum[(1/((((i j k+i+k))^2)(((i j k+i j+i+k+1))^2))),{i,∞},{j,∞},{k,∞}]

Table[((1.` (2.` (((i j k+i j+i+k+1.`))^2) i k (i j k+i+k)+2.` (i j k+i j+i+k+1.`) (((i j k+i+k))^2) (i+i k)))/((((((i j k+i+k))^2) (((i j k+i j+i+k+1.`))^2)))^2))/.{k->1.,j->RandomInteger[100],i->RandomInteger[100]},{20}]//Chop

Table[D[((1.` (2.` (((i j k+i j+i+k+1.`))^2) i k (i j k+i+k)+2.` (i j k+i j+i+k+1.`) (((i j k+i+k))^2) (i+i k)))/((((((i j k+i+k))^2) (((i j k+i j+i+k+1.`))^2)))^2)),k]/.{k->1.,j->RandomInteger[100],i->RandomInteger[100]},{20}]//Chop

Simplify[D[(1/((((i j k+i+k))^2)(((i j k+i j+i+k+1))^2))),k]/.k->1]

N[Sum[(1/((((i j k+i+k))^2)(((i j k+i j+i+k+1))^2))),{i,20},{j,20},{k,20}],20]