CMRB=
∞
∑
x=1
x
(-1)
(
1/x
x
-1)
CMRB=
lim
k∞
∞
∑
x=1
x
(-1)

1
x
x
-1k!-k
1
x
x
Γ(k)-Γk,
log(x)
x

k(k-1)!
CMRB=
lim
k∞
∞
∑
x=1
x
(-1)
(
1/x
x
-1)Γk,
log(x)
x

Γ(k)
It seems that for optimal performance in computing d digits, use the following; later listed is better.
CMRB=
∞
∑
x=1
x
(-1)

1
x
x
-1k!-k
1
x
x
Γ(k)-Γk,
log(x)
x

k(k-1)!
/.k
d
2
.
CMRB=
∞
∑
x=1
x
(-1)
(
1/x
x
-1)Γk,
log(x)
x

Γ(k)
/.k2d
CMRB=
∞
∑
x=1
x
(-1)
(
1/x
x
-1)Γk,
log(x)
x

Γ(k)
/.k
7d
8
CMRB=
∞
∑
x=1
x
(-1)
-
1
x
-1
x
log(x)
Γ(k)
+
1/x
x
-1/.k
d
2
​​​​
In[]:=
ClearSystemCache[]
In[]:=
f[x_]=(-1)^x(x^(1/x)-1);
In[]:=
Timing[CMRB=NSum[f[x],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000];]
Out[]=
{0.640625,Null}
In[]:=
Timing[CMRB-NSum[(((-1)^x*((-1+x^x^(-1))*k!-k*x^x^(-1)*(Gamma[k]-Gamma[k,Log[x]/x])))/(k*(-1+k)!))/.k500,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000]]
Out[]=
{1.8125,-3.×
-998
10
}
In[]:=
Timing[CMRB-NSum[(((-1)^x*((-1+x^x^(-1))*k!-k*x^x^(-1)*(Gamma[k]-Gamma[k,Log[x]/x])))/(k*(-1+k)!))/.k1000,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000]]
Out[]=
{7.5625,0.×
-1999
10
}
In[]:=
Assuming[k∈Integers&&k>0,FullSimplify[(((-1)^x*((-1+x^x^(-1))*k!-k*x^x^(-1)*(Gamma[k]-Gamma[k,Log[x]/x])))/(k*(-1+k)!))]]
Out[]=
x
(-1)
-1+
1
x
x
Gammak,
Log[x]
x

Gamma[k]
CMRB=
∞
∑
x=1
x
(-1)
-1+
1
x
x
Gammak,
Log[x]
x

Gamma[k]
/.k2d
In[]:=
ClearSystemCache[]
In[]:=
f[x_]=(-1)^x(x^(1/x)-1);
In[]:=
Timing[CMRB=NSum[f[x],{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000];]
Out[]=
{0.6875,Null}
In[]:=
ClearSystemCache[]
In[]:=
Timing[CMRB-NSum[((f[x]*Gamma[k,Log[x]/x])/Gamma[k])/.k2000,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000]]
Out[]=
{1.73438,0.×
-999
10
}
In[]:=
ClearSystemCache[]
In[]:=
Timing[CMRB-NSum[((f[x]*Gamma[k,Log[x]/x])/Gamma[k])/.k4000,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000]]
Out[]=
{7.04688,3.×
-998
10
}
CMRB=
∞
∑
x=1
x
(-1)
-1+
1
x
x
Gammak,
Log[x]
x

Gamma[k]
/.k
7
8
d
ClearSystemCache[]
In[]:=
Timing[CMRB-NSum[((f[x]*Gamma[k,Log[x]/x])/Gamma[k])/.k3500,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision2000]]
Out[]=
{6.21875,0.×
-1999
10
}
CMRB=
∞
∑
x=1
x
(-1)
-
1
x
-1
x
log(x)
Γ(k)
+
1/x
x
-1/.k
d
2
In[]:=
Timing[CMRB-NSum[(-1)^x*(-1+x^x^(-1)-(x^(-1+x^(-1))*Log[x])/Gamma[k])/.k500,{x,1,Infinity},Method"AlternatingSigns",WorkingPrecision1000]]
Out[]=
{0.984375,0.×
-999
10
}