In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision->100,Method->"AlternatingSigns"];a=-InfinityI;b=InfinityI;​​g[x_]=x^(1/x);(v=t/(1+t+tI);​​Print[CMRB-(-I/2NIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,a,b},WorkingPrecision->100])]);Clear[a,b]
-9.3472×
-94
10
In[]:=
CMRB-(-INIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,0,InfinityI},WorkingPrecision->100]+INIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,I,InfinityI},WorkingPrecision->100])
Out[]=
-9.3472×
-94
10
CMRB=-
∞
∫

Re
-v
v
csc
π
v

2
t
t-
∞
∫
0
Re
-v
v
csc
π
v

2
t
t
In[]:=
CMRB-(-INIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,0,I},WorkingPrecision->100])
Out[]=
-9.3472×
-94
10
CMRB=-

∫
0
Re
-v
v
csc
π
v

2
t
t
CMRB=-
∞
∫

Re
-v
v
csc
π
v

2
t
t-
∞
∫
0
Re
-v
v
csc
π
v

2
t
t=-

∫
0
Re
-v
v
csc
π
v

2
t
t
In[]:=
1/-I
Out[]=

g(x_)=
1/x
x
;v=
t
t+t+1
;​​CMRB=
∞
∫

Re
-v
v
csc
π
v

2
t
t-
∞
∫
0
Re
-v
v
csc
π
v

2
t
t=

∫
0
Re
-v
v
csc
π
v

2
t
t