f(x)=
x
(-1)
1-
1
x+1
(x+1)

CMRB=
∞
∑
n=0
f(n)=
∞
∑
n=1
f(n)=
∞
∑
n=1
h(n)

∞
∫
0
(f(t)-f(-t))
(
2πt

-1)
t+

N∞
2N+1
∫
1
πt

g(t)t=CMRB
In[]:=
f(x_)=
x
(-1)
1-
1
x+1
(x+1)
;h(x_)=
x
(-1)
(
1/x
x
-1);g(x_)=
1/x
x
;
In[]:=
NIntegrate[E^(Pi*I*t)*g[t],{t,1,10001},WorkingPrecision->10]
Out[]=
0.07079022316-0.04706930965
In[]:=
MKB=-INIntegrate[g[1+It]/E^(Pit),{t,0,Infinity},WorkingPrecision100,MaxRecursion20]-I/Pi
Out[]=
0.0707760393115288035395280218302820013657546962033630275831727881636184572643820365808318812661772382-0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787
In[]:=
CMRB=NSum[h[n],{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision100];
In[]:=
term=NIntegrate[((I(f[It]-f[-It])))/(E^(2Pit)-1),{t,0,Infinity},WorkingPrecision100,Method"GlobalAdaptive"]
Out[]=
0.11708360315053831670898991222399122869014839869677575858883189592585877430027817712246477316693025869+0.04738061707035078610720940650260367857315289969317363933196100090256586758807049779050462314770913485
In[]:=
(term+MKB-CMRB)
Out[]=
9.3472×
-94
10
+0.×
-101
10
