On5/17/2022MarvinRayBurnsfoundthefollowingapproximationforVCusingCMRB,W(1)and
π
E
.​​Viswanath'sconstant(VC)​​theMRBconstant(CMRB),​​ProductLog[1](W(1))​​Gelfond'sconstant(
π
E
).
VC
6
11
7
-W(1)
-CMRB
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",WorkingPrecision21]
Out[]=
0.18785964246234485656
Here are the errors in two different approximations.
In[]:=
VC
6(11/7-ProductLog[1])
-CMRB​​CMRB-
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/.SolveReduce
v
6(11/7-ProductLog[1])
-CMRB
CMRB-
5
6
v
56
==E^Pi,v[[6]],v
Out[]=
{0.×
-12
10
}