The terms and the upper limit point of their partial sums of the MRB constant (CMRB) series:
The terms and the upper limit point of their partial sums of the MRB constant (CMRB) series:
In[]:=
CMRB=1/2+NSum[,{k,1,Infinity},WorkingPrecision30,Method"AlternatingSigns"]
k
(-1)
1/k
k
Out[]=
0.1878596424620671202361147732
In[]:=
Assuming[q<1,Sum[q^k,{k,1,Infinity}]]
Out[]=
-
q
-1+q
Considerthegeometricsum,p=-,where.
q
-1+q
k
q
log-
x+π
log(x)
2
x
πx
log(q)
In[]:=
p=FullSimplify-/.Solve[q^k==Log[-E^(I*Pi*x)+E^(x*(I*Pi+Log[x]/x^2))]/Log[q],q]
q
-1+q
p=-1+
1
1-
1
k
ProductLogkLog-1+
πx
1
x
x
Below, by fixing the value of k, the plot of the terms of the geometric sum of the CMRB looks similar that of the partial sums of the CMRB series.
Below, by fixing the value of k, the plot of the terms of the geometric sum of the CMRB looks similar that of the partial sums of the CMRB series.
In[]:=
Show[ListPlot[l=Accumulate[Table[,{k,100}]],JoinedTrue,PlotStyleRed,AxesLabel{n,}],Plot[CMRB-1,{n,1,100},PlotStyleBlue],Plot[CMRB,{n,1,100},PlotStyleBlue]]
k
(-1)
1/k
k
s
n
In[]:=
Table[y[n]=ReImPlot[p/.k->n,{x,1,100},PlotStyleGreen],{n,1/100,1,1/100}];
In[]:=
Show[{Table[y[n],{n,1/100,1,1/100}],Plot[CMRB,{n,1,100},PlotStyleYellow],Plot[CMRB-1,{n,1,100},PlotStyleYellow]},PlotRange->{-1,1}]
By fixing the value of k, the real and imaginary parts of this function, p(x)=-1+11-1kProductLogkLogπx-1+1xx, go close to the upper and lower limit points of the CMRB series.
By fixing the value of k, the real and imaginary parts of this function, , go close to the upper and lower limit points of the CMRB series.
p(x)=-1+
1
1-
1
k
ProductLogkLog-1+
πx
1
x
x
In[]:=
Table[y[n]=ReImPlot[p/.k->n,{x,1,100},PlotStyleGreen],{n,1/100,1,1/100}];
In[]:=
p=FullSimplify-/.Solve[q^k==Log[-E^(I*Pi*x)+E^(x*(I*Pi+Log[x]/x^2))]/Log[q],q]
q
-1+q
In[]:=
p/.k->1
Out[]=
-1+
1
1-
ProductLogLog-1+
πx
1
x
x
In[]:=
Show[{y[100/100],Plot[CMRB-1,{n,1,100},PlotStyleYellow]},PlotRange->{-1,1}]
p(x)=-1+
1
1-
ProductLogLog-1+
πx
1
x
x
In[]:=
ShowReImPlot-1+,{x,1,100},PlotStyleBlack,Plot[CMRB,{x,1,100},PlotStyleYellow],Plot[CMRB-1,{x,1,100},PlotStyleYellow]
1
1-
ProductLogLog-1+
πx
1
x
x
In[]:=
ShowReImPlot-1+,{x,1,1000},Plot[CMRB,{x,1,1000},PlotStyleRed],Plot[CMRB-1,{x,1,1000},PlotStyleBlack]
1
1-
ProductLogLog-1+
πx
1
x
x
In[]:=
ShowReImPlot-1+,{x,1,10000},Plot[CMRB,{x,1,10000},PlotStyleRed],Plot[CMRB-1,{x,1,10000},PlotStyleBlack]
1
1-
ProductLogLog-1+
πx
1
x
x
However, the details of the geometric sum are rather intricate.
However, the details of the geometric sum are rather intricate.