f[x_]=
πx

1-
1
1+x
(1+x)
;
CMRB=NSum[f[n],{n,0,Infinity},WorkingPrecision30,Method"AlternatingSigns"];
M2=NIntegrate[f[t],{t,0,InfinityI},WorkingPrecision50];
In[]:=
part=NIntegrate(Im[2f[(-t)]]+(f[(-t)]-f[(t)]))(-1+
-2πt

),{t,​​0,InfinityI},WorkingPrecision50;
In[]:=
CMRB(1-I)-(M2-part)
Out[]=
6.103779×
-23
10
-6.103779×
-23
10

​​