Mission, integrate the terms of the Taylor series and find a pattern for their sum.
In[]:=
l2=Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,3}]];
In[]:=
l3=NIntegrate[l2,{x,1,InfinityI}]
Out[]=
0.070776-0.0473806
In[]:=
l2
Out[]=
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
In[]:=
Integrate[l2,{x,1,InfinityI}]
Out[]=
MeijerG[{{},{1,1}},{{0,0,0},{}},-π]-πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]-MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]
2
π
The above is the pattern for the sum. What does it look like?
In[]:=
l2=Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,3}]];
The following are the same!
In[]:=
l3=NIntegrate[l2,{x,1,Infinity}]
Out[]=
0.0706678-0.0474642
In[]:=
l3=NIntegrate[l2,{x,1,InfinityI}]
Out[]=
0.0706678-0.0474642
In[]:=
l2
Out[]=
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
In[]:=
Integrate[l2,{x,1,Infinity}]
Out[]=
1
3840
4
EulerGamma
2
π
6
π
3
EulerGamma
2
π
5
π
4
π
5
2
2
π
4
2
Log[π]
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
π
4
2
Log[π]
2
EulerGamma
4
π
3
π
2
π
2
Log[π]
1
2
1
2
3
2
3
2
3
2
2
π
4
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
3
2
2
π
4
2
Log[π]
3
Log[π]
3
π
5
2
2
π
4
2
Log[π]
3
Log[π]
2
π
3
2
2
π
4
2
Log[π]
3
Log[π]
4
Log[π]
5
π
4
π
2
Log[π]
3
π
2
Log[π]
2
π
2
Log[π]
3
Log[π]
In[]:=
l2=Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,6}]];
In[]:=
l3=NIntegrate[l2,{x,1,InfinityI}]
Out[]=
0.070776-0.0473807
In[]:=
l2
Out[]=
πx
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
4
Log[x]
24
4
x
5
Log[x]
120
5
x
6
Log[x]
720
6
x
In[]:=
Integrate[l2,{x,1,InfinityI}]
Out[]=
MeijerG[{{},{1,1}},{{0,0,0},{}},-π]-πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]-MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]+MeijerG[{{},{1,1,1,1,1}},{{-3,0,0,0,0,0},{}},-π]+MeijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-π]-MeijerG[{{},{1,1,1,1,1,1,1}},{{-5,0,0,0,0,0,0,0},{}},-π]
2
π
3
π
4
π
5
π
Here is more of the pattern.
MeijerG[{{},{1,1}},{{0,0,0},{}},-π]-πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]-MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]Theaboveisthepatternforthesumand.Whatdoesitlooklike?MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]+MeijerG[{{},{1,1,1,1,1}},{{-3,0,0,0,0,0},{}},-π]+MeijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-π]-MeijerG[{{},{1,1,1,1,1,1,1}},{{-5,0,0,0,0,0,0,0},{}},-π]Thereismoreofthepattern.
2
π
MeijerG
[
{{},{1,1}},{{0,0,0},{}},-π
]
-
πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]
-2
π
3
π
4
π
5
π
M2=x==-1+x==f/.f=MeijerG[{{},{1,1...1+ntimes}},{{1-n,0,0,...1+ntimes},{}},-π]
n∞
2n+1
∫
1
πx
1
x
x
∞
∫
1
πx
1
x
x
∞
∑
n=1
1-n
π
In[]:=
N[MeijerG[{{},{1,1}},{{0,0,0},{}},-Iπ]-IπMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-Iπ]-π^2MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-Iπ]+Iπ^3MeijerG[{{},{1,1,1,1,1}},{{-3,0,0,0,0,0},{}},-Iπ]+π^4MeijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-Iπ]-Iπ^5MeijerG[{{},{1,1,1,1,1,1,1}},{{-5,0,0,0,0,0,0,0},{}},-Iπ]]
Out[]=
0.070776-0.0473807
6
∑
n=1
n
(π)
Try to get it in this Traditional form
In[]:=
0.0430039148058859653072069391181