On5/17/2022MarvinRayBurnsfoundthefollowingapproximationforVCusingCMRB,W(1)and.Viswanath'sconstant(VC)theMRBconstant(CMRB),ProductLog[1](W(1))Gelfond'sconstant().
π
E
π
E
VC
6-W(1)
11
7
CMRB-
5
6
VC
56
π
E
In[]:=
VC=1.1319882487943`12
Out[]=
1.13198824879
In[]:=
CMRB=NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision21]
Out[]=
0.18785964246234485656
Here are the errors in two different approximations.
In[]:=
VC
6(11/7-ProductLog[1])
5
6
VC
56
Out[]=
3.4164×
-8
10
Out[]=
1.47×
-9
10
How close of an approximation to VC, does the ratio of those two approximations give?
In[]:=
VC-v/.SolveReduce-CMRB==E^Pi,v[[6]],v
v
6(11/7-ProductLog[1])
CMRB-
5
6
v
56
Out[]=
{0.×}
-12
10