In[]:=
With[​​ {area=NIntegrate[sumTerm[t],{t,0,1}]},​​ Plot[​​ sumTerm[t],​​ {t,0,1},​​ Filling->Axis,​​ Frame->True,​​ FrameLabel->{"t","sumTerm[t]"},​​ PlotLabel->Row[{"Area = ",NumberForm[area,10]}]​​ ]​​ ]
NSum
:The precision of the argument function ​
n
(-1)
Log[n]
0.0000204286
n
-1
n
​ is less than WorkingPrecision (​210.​).
Out[]=
In[]:=
expr=Im[((1-I(1-t)/t)^(1/(1-I(1-t)/t))Csch[Pi(-1+t)/t])/t^2];​​​​area=NIntegrate[expr,{t,0,1}];​​​​Plot[​​expr,​​{t,0,1},​​Filling->Axis,​​PlotRange->All,​​PlotPoints->100,​​MaxRecursion->6,​​Frame->True,​​FrameLabel->{"t","Im[...] / t^2"},​​PlotLabel->Row[{"Area = ",NumberForm[area,10]}]​​]
General
:Csch[-310704.] is too small to represent as a normalized machine number; precision may be lost.
Out[]=
In[]:=
CMRB=
∞
∑
n=1
n
(-1)
-1+
1
n
n
;
In[]:=
∞
∫
1
Log[n]
n
∫
0
u

uIm[Sin[Pin]]n//Re//N//Chop
Out[]=
0.18786
In[]:=
NIntegrate[​​-I((1+It)^(1/(1+It))-1)/Sinh[Pit],​​{t,0,1,Infinity}​​]
Out[]=
0.18786-0.101409
In[]:=
Plot[​​Im[I((1+It)^(1/(1+It))-1)/Sinh[Pit]],​​{t,0,Infinity},​​PlotRange->Full,​​AxesLabel->{"t","value"},​​PlotLabel->Row[{​​"Imaginary part of ",​​TraditionalForm[I((1+It)^(1/(1+It))-1)/Sinh[Pit]]​​}],​​LabelStyle->Directive[14],​​PlotTheme->"Scientific"​​]
Out[]=
}
In[]:=
f[n_]:=
2
Log[n]
;2CMRB-
∞
∑
n=1
f'[n]
∫
0
u/2-πn

u//N
In[]:=
f[n_]:=
2
Log[n]
;
∞
∑
n=1
f'[n]
∫
0
u/2-πn

u
Out[]=
∞
∑
n=1
2
-nπ

-1+
1
n
n

In[]:=
f[n_]:=
2
Log[n]
;1/2
∞
∑
n=1
f'[n]
∫
0
u/2-πn

u//N
Out[]=
0.18786
In[]:=
DiscretePlot[​​Sum[(-1)^n(n^(1/n)-1),{n,1,m}],​​{m,1,20},​​AxesLabel->{"m","partial sum"},​​PlotLabel->"Partial sums of alternating series",​​LabelStyle->Directive[14],​​PlotTheme->"Scientific"​​]
Out[]=
In[]:=
CMRB-
∞
∑
n=1
n
(-1)
-1
n
1
∫
0
u/n
n
log(n)u
Out[]=
0
In[]:=
DiscretePlot[​​(-1)^n(n-1)/(n^2Log[n]^2),​​{n,2,20},​​AxesLabel->{"n","summand"},​​PlotLabel->"Summand of the series",​​LabelStyle->Directive[14],​​PlotTheme->"Scientific"​​]