∞
∫
1
x
(-1)
(
1/x
x
-1)x​​=
∞
∫
0
πx

1-
1
x+1
(x+1)
x​​=
∞
∫
0
πx

1-
1
x+1
(x+1)
x​​=
∞
∫
1
1/x
x
-1
cos(πx)-sin(πx)
x​​
∞
∑
x=1
1/x
x
-1
cos(πx)-sin(πx)
​​=
∞
∑
x=1
x
(-1)
(
1/x
x
-1).
In[]:=
NIntegrate[
x
(-1)
(
1/x
x
-1),{x,1,Infinity},WorkingPrecision10]
Out[]=
0.07078110711-0.04737887018
In[]:=
NIntegrate[E^(IPix)(1-(x+1)^(1/(x+1))),{x,0,Infinity},WorkingPrecision10]
Out[]=
0.07077603940-0.04738061718
In[]:=
NIntegrate[E^(IPix)(1-(x+1)^(1/(x+1))),{x,0,InfinityI},WorkingPrecision30,Method"Trapezoidal"]
Out[]=
0.0707760393115288035395280218303-0.0473806170703507861072094065026
In[]:=
NIntegrate[(t^(1/t)-1)/(Cos[Pit]-ISin[Pit]),{t,1,InfinityI},WorkingPrecision30,Method"Trapezoidal"]
Out[]=
0.0707760393115288035395280218303-0.0473806170703507861072094065026
In[]:=
NSum[(n^(1/n)-1)/(Cos[Pin]-ISin[Pin]),{n,1,Infinity}]
Out[]=
0.18786
In[]:=
NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity}]
Out[]=
0.18786