In[]:=
f[x_]=(-1)^x(x^(1/x)-1);
In[]:=
CMRB=NSum[f[x],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision100]
Out[]=
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278
In[]:=
l=TableSum
1
(k-1)!
k-1
x
(-1)^xLog[x]Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{x,1,Infinity},{k,1,10}//Short//TableForm
Out[]//TableForm=
0,
1
2
(2EulerGammaLog[2]-
2
Log[2]
),6,
1
4800
,
82+1
48771072000

In[]:=
(la=Expand[Table[l[[n+1]]-l[[n]],{n,1,9}]])//TableForm
Out[]//TableForm=
EulerGammaLog[2]-
2
Log[2]
2
-
1
12
EulerGamma
2
π
Log[2]-
1
24
2
π
2
Log[2]
+
2
π
Log[2]Log[Glaisher]-
1
12
2
π
Log[2]Log[π]-
′′
Zeta
[2]
4
1
24
3
Log[2]
Zeta[3]-
1
8
2
Log[2]
′
Zeta
[3]+
1
24
Log[8]
′′
Zeta
[3]+
1
8
(3)
Zeta
[3]
4
π
4
Log[2]
17280
-
1
48
3
Log[2]
′
Zeta
[4]+
1
32
2
Log[2]
′′
Zeta
[4]-
1
48
Log[2]
(3)
Zeta
[4]-
7
192
(4)
Zeta
[4]
5
Log[2]
Zeta[5]
1920
-
1
384
4
Log[2]
′
Zeta
[5]+
1
192
3
Log[2]
′′
Zeta
[5]-
1
192
2
Log[2]
(3)
Zeta
[5]+
Log[32]
(4)
Zeta
[5]
1920
+
1
128
(5)
Zeta
[5]
6
π
6
Log[2]
21772800
-
5
Log[2]
′
Zeta
[6]
3840
+
4
Log[2]
′′
Zeta
[6]
1536
-
3
Log[2]
(3)
Zeta
[6]
1152
+
2
Log[2]
(4)
Zeta
[6]
1536
-
Log[2]
(5)
Zeta
[6]
3840
-
31
(6)
Zeta
[6]
23040
7
Log[2]
Zeta[7]
322560
-
6
Log[2]
′
Zeta
[7]
46080
+
5
Log[2]
′′
Zeta
[7]
15360
-
4
Log[2]
(3)
Zeta
[7]
9216
+
3
Log[2]
(4)
Zeta
[7]
9216
-
2
Log[2]
(5)
Zeta
[7]
15360
+
Log[2]
(6)
Zeta
[7]
46080
+
(7)
Zeta
[7]
5120
8
π
8
Log[2]
48771072000
-
7
Log[2]
′
Zeta
[8]
645120
+
6
Log[2]
′′
Zeta
[8]
184320
-
5
Log[2]
(3)
Zeta
[8]
92160
+
4
Log[2]
(4)
Zeta
[8]
73728
-
3
Log[2]
(5)
Zeta
[8]
92160
+
2
Log[2]
(6)
Zeta
[8]
184320
-
Log[2]
(7)
Zeta
[8]
645120
-
127
(8)
Zeta
[8]
5160960
9
Log[2]
Zeta[9]
92897280
-
8
Log[2]
′
Zeta
[9]
10321920
+
7
Log[2]
′′
Zeta
[9]
2580480
-
6
Log[2]
(3)
Zeta
[9]
1105920
+
5
Log[2]
(4)
Zeta
[9]
737280
-
4
Log[2]
(5)
Zeta
[9]
737280
+
3
Log[2]
(6)
Zeta
[9]
1105920
-
2
Log[2]
(7)
Zeta
[9]
2580480
+
Log[2]
(8)
Zeta
[9]
10321920
+
17
(9)
Zeta
[9]
6193152
In[]:=
(ll=Table[(Sum[(-1)^xLog[x]^n/(n!x^n),{x,1,Infinity}]),{n,​​1,9}])//Short//TableForm
Out[]//TableForm=

1
2
(2EulerGammaLog[2]-
2
Log[2]
),
1
24
(
2
π
2
1
-1-61),6,
1
92897280

In[]:=
Expand[Table[ll[[n]]-l[[n+1]]+l[[n]],{n,1,9}]]
Out[]=
{0,0,0,0,0,0,0,0,0}
In[]:=
Expand[Table[ll[[n]]-la[[n]],{n,1,9}]]
Out[]=
{0,0,0,0,0,0,0,0,0}
In[]:=
CMRB-(l)//N
Out[]=
{0.18786,0.0279907,0.00280295,0.000207355,0.0000118838,5.3695×
-7
10
,1.87833×
-8
10
,4.54955×
-10
10
,2.79824×
-12
10
,-4.4772×
-13
10
}
In[]:=
(ll)//N
Out[]=
{0.159869,0.0251878,0.0025956,0.000195471,0.0000113469,5.18166×
-7
10
,1.83284×
-8
10
,4.52157×
-10
10
,3.24596×
-12
10
}
In[]:=
N[CMRB-(ll+l[[1;;9]]),30]
Out[]=
{0.0279907387196361484915700637294,0.00280295020690991487554302477903,0.000207354973003898926467754641480,0.0000118838220054139752268887360275,5.36949668548612943348287695556×
-7
10
,1.87833475995791223751662833641×
-8
10
,4.54955477824640390711570422552×
-10
10
,2.79823510102935621930983118607×
-12
10
,-4.47720121506293658470300505848×
-13
10
}
In[]:=
Chop[%]
Out[]=
{0.0279907387196361484915700637294,0.00280295020690991487554302477903,0.000207354973003898926467754641480,0.0000118838220054139752268887360275,5.36949668548612943348287695556×
-7
10
,1.87833475995791223751662833641×
-8
10
,4.54955477824640390711570422552×
-10
10
,0,0}