In[]:=
f[x_]=Exp[IPix](x^(1/x)-1);
In[]:=
CMRB=NSum[f[x],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision100]
Out[]=
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278
In[]:=
Table[Sum[Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,n}]][[1;;2]],{x,1,Infinity}],{n,1,2}]
Out[]=
-Log[2],(24EulerGammaLog[2]-2EulerGammaLog[2]-12-+24Log[2]Log[Glaisher]-2Log[2]Log[π]-6[2])
1
24
2
π
2
Log[2]
2
π
2
Log[2]
2
π
2
π
′′
Zeta
In[]:=
Table[Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,n}]][[1;;2]],{n,1,2}]
Out[]=
,+
πx
x
πx
Log[x]
x
2
Log[x]
2
2
x
In[]:=
p=Table[Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,n}]][[1;;2]],{n,1,3}]
Out[]=
,+,++
πx
x
πx
Log[x]
x
2
Log[x]
2
2
x
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
In[]:=
FullSimplify[p]
Out[]=
,Log[x](2x+Log[x]),Log[x](6+3xLog[x]+)
πx
x
πx
2
2
x
πx
2
x
2
Log[x]
6
3
x
In[]:=
p=Table[Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,n}]][[1;;2]],{n,1,4}]
Out[]=
,+,++,+++
πx
x
πx
Log[x]
x
2
Log[x]
2
2
x
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
4
Log[x]
24
4
x
In[]:=
q=FullSimplify[p]
Out[]=
,Log[x](2x+Log[x]),Log[x](6+3xLog[x]+),Log[x](24+12Log[x]+4x+)
πx
x
πx
2
2
x
πx
2
x
2
Log[x]
6
3
x
πx
3
x
2
x
2
Log[x]
3
Log[x]
24
4
x
In[]:=
1/2-N[Sum[q[[2]],{x,1,10}]]
Out[]=
0.190557
In[]:=
1/2-N[Sum[q[[3]],{x,1,10}]]
Out[]=
0.18703
In[]:=
p=Table[Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,n}]][[1;;2]],{n,1,6}]
Out[]=
,+,++,+++,++++,+++++
πx
x
πx
Log[x]
x
2
Log[x]
2
2
x
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
4
Log[x]
24
4
x
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
4
Log[x]
24
4
x
5
Log[x]
120
5
x
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
4
Log[x]
24
4
x
5
Log[x]
120
5
x
6
Log[x]
720
6
x
In[]:=
q=FullSimplify[p]
Out[]=
,Log[x](2x+Log[x]),Log[x](6+3xLog[x]+),Log[x](24+12Log[x]+4x+),Log[x](120+60Log[x]+20+5x+),Log[x](720+360Log[x]+120+30+6x+)
πx
x
πx
2
2
x
πx
2
x
2
Log[x]
6
3
x
πx
3
x
2
x
2
Log[x]
3
Log[x]
24
4
x
1
120
5
x
πx
4
x
3
x
2
x
2
Log[x]
3
Log[x]
4
Log[x]
1
720
6
x
πx
5
x
4
x
3
x
2
Log[x]
2
x
3
Log[x]
4
Log[x]
5
Log[x]
Out[]=
,Log[x](2x+Log[x]),Log[x](6+3xLog[x]+),Log[x](24+12Log[x]+4x+)Log[x](120+60Log[x]+20+5xLog[x](720+360Log[x]+120+30+6x+)
πx
x
πx
2
2
x
πx
2
x
2
Log[x]
6
3
x
πx
3
x
2
x
2
Log[x]
3
Log[x]
24
4
x
,
1
120
5
x
πx
4
x
3
x
2
x
2
Log[x]
3
Log[x]
+)
4
Log[x]
,
1
720
6
x
πx
5
x
4
x
3
x
2
Log[x]
2
x
3
Log[x]
4
Log[x]
5
Log[x]
In[]:=
FactorLog[x]Sum[7!/(n)!x^(6-n)Log[x]^(n-1)/7,{n,1,6}]
πx
6!
6
x
Out[]=
1
720
6
x
πx
5
x
4
x
3
x
2
Log[x]
2
x
3
Log[x]
4
Log[x]
5
Log[x]
In[]:=
TableFactorLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{k,2,7}//TableForm
1
(k-1)!
k-1
x
πx
Out[]//TableForm=
πx x |
πx 2 2 x |
πx 2 x 2 Log[x] 6 3 x |
πx 3 x 2 x 2 Log[x] 3 Log[x] 24 4 x |
πx 4 x 3 x 2 x 2 Log[x] 3 Log[x] 4 Log[x] 120 5 x |
πx 5 x 4 x 3 x 2 Log[x] 2 x 3 Log[x] 4 Log[x] 5 Log[x] 720 6 x |
In[]:=
PrependTableFactorLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{k,3,7},-q
1
(k-1)!
k-1
x
πx
πx
x
Out[]=
{{0,0,0,0,0,0}}
In[]:=
FactorLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}]/.k7
1
(k-1)!
k-1
x
πx
Out[]=
-5040-7Gamma7,
πx
1
x
x
Log[x]
x
5040
In[]:=
CMRB-NSumFactorLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}]/.k7,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision30
1
(k-1)!
k-1
x
πx
Out[]=
1.87833475995791223752×
-8
10
In[]:=
CMRB-TableNSumLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision30,{k,1,20}
1
(k-1)!
k-1
x
πx
Out[]=
$Aborted
In[]:=
CMRB-TableNSum(-1)^xLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision30,{k,1,20}
1
(k-1)!
k-1
x
Out[]=
{0.18785964246206712024851793405,0.02799073871963614849150875859,0.00280295020690991487548198694,0.00020735497300389892640671687,0.00001188382200541397516585095,5.3694966854861288231050×,1.878334759957906133738×,4.5495547782457935292×,2.79823510096831843×,-4.4772012156733145×,-3.244467539586037×,-1.50314588913226×,-5.620271131815×,-1.81795935390×,-5.258826620×,-1.44212338×,-9.448374×,-6.179033×,-6.105363×,-6.103810×}
-7
10
-8
10
-10
10
-12
10
-13
10
-14
10
-15
10
-17
10
-18
10
-20
10
-21
10
-23
10
-23
10
-23
10
-23
10
In[]:=
CMRB-TableNSum(-1)^xLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision50,{k,1,20}
1
(k-1)!
k-1
x
Out[]=
{0.1878596424620671202485179340542732300559030949001,0.0279907387196361484915700637293566595596808436573,0.0028029502069099148755430247790316247030858833367,0.0002073549730038989264677546414802829607336299980,0.0000118838220054139752268887360274675549814454632,5.369496685486129433482876955560049589146459×,1.87833475995791223751662833640792348872811×,4.549554778246403907115704225522396253991×,2.7982351010293562193098311860667233284×,-4.477201215062936584703005058479086756×,-3.24446753348225839015066851558692544×,-1.5031458280944732567295317498796235×,-5.62026502803637919956481092217063×,-1.8178983161060950034204221797312×,-5.25272284140188192394382877646×,-1.3810855952693614750955928226×,-3.34459529586118379026076925×,-7.525382081630412820870630×,-1.58362528986859625251466×,-3.133328802739467959461×}
-7
10
-8
10
-10
10
-12
10
-13
10
-14
10
-15
10
-17
10
-18
10
-20
10
-21
10
-23
10
-25
10
-26
10
-28
10
In[]:=
CMRB-Table[Timing[NSum[((-1)^xLog[x]Sum[k!/(n)!x^(k-1-n)N[Log[x],50]^(n-1)/k,{n,1,k-1}])/((k-1)!x^(k-1)),{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision50]],{k,1,20}]