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]-
2
π
MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-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!
∞
∫
1
(
1/x
x
-1)exp(πx)x
∞
∫
1
(
1/x
x
-1)exp(πx)x
In[]:=
FullSimplify[N[NIntegrate[Exp[IPix](x^(1/x)-1),{x,1,Infinity},WorkingPrecision160,MaxRecursion300],60]-NIntegrate[Exp[IPix](x^(1/x)-1),{x,1,InfinityI},WorkingPrecision60]]
Out[]=
0.×
-61
10
+0.×
-61
10

In[]:=
NIntegrate[l2,{x,1,Infinity}]-NIntegrate[l2,{x,1,InfinityI}]
NIntegrate
:DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.
NIntegrate
:DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.
Out[]=
-1.04802×
-11
10
-1.62902×
-9
10

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
-80
4
EulerGamma
2
π
+3
6
π
+160
3
EulerGamma
π(-4+
2
π
+π(3-2Log[π]))+20
5
π
(-3+2Log[π])+10
4
π
6+HypergeometricPFQ{1,1,1,1,1},2,2,2,2,
5
2
,3,-
2
π
4
-12Log[π]+4
2
Log[π]
+2401+16HypergeometricPFQ-
1
2
,-
1
2
,-
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,-
2
π
4
+8
2
Log[π]
+40
2
EulerGamma
(48+
4
π
-48π(-1+Log[π])+6
3
π
(-3+2Log[π])-6
2
π
(11-6Log[π]+2
2
Log[π]
))+640π6+6HypergeometricPFQ
1
2
,
1
2
,
3
2
,
3
2
,
3
2
,-
2
π
4
+6HypergeometricPFQ-
1
2
,-
1
2
,-
1
2
,-
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
3
2
,-
2
π
4
-9Log[π]+3
2
Log[π]
-
3
Log[π]
-2Zeta[3]+40
3
π
-49+2HypergeometricPFQ{1,1,1,1},2,2,2,2,
5
2
,-
2
π
4
+46Log[π]-18
2
Log[π]
+4
3
Log[π]
+8Zeta[3]-40
2
π
145+12HypergeometricPFQ{1,1,1},
3
2
,2,2,2,-
2
π
4
+66
2
Log[π]
-12
3
Log[π]
+2
4
Log[π]
-24Zeta[3]+2Log[π](-69+8Zeta[3])+40EulerGamma(
5
π
+96Log[π]+
4
π
(-3+2Log[π])-48π(3-2Log[π]+
2
Log[π]
)+2
3
π
(23-18Log[π]+6
2
Log[π]
)-2
2
π
(-69+66Log[π]-18
2
Log[π]
+4
3
Log[π]
+8Zeta[3]))
That is too messy!
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]-
2
π
MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]+
3
π
MeijerG[{{},{1,1,1,1,1}},{{-3,0,0,0,0,0},{}},-π]+
4
π
MeijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-π]-
5
π
MeijerG[{{},{1,1,1,1,1,1,1}},{{-5,0,0,0,0,0,0,0},{}},-π]
Above is more of the pattern.
MeijerG[{{},{1,1}},{{0,0,0},{}},-π]-πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]-
2
π
MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]​​Theaboveisthepatternforthesumand.Whatdoesitlooklike?​​
MeijerG
[
{{},{1,1}},{{0,0,0},{}},-π
]
-
πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]
-
2
π
MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]+
3
π
MeijerG[{{},{1,1,1,1,1}},{{-3,0,0,0,0,0},{}},-π]+
4
π
MeijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-π]-
5
π
MeijerG[{{},{1,1,1,1,1,1,1}},{{-5,0,0,0,0,0,0,0},{}},-π]​​Thereismoreofthepattern.
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)
MeijerG[{{},{1,1}},{{n,n,n},{}},-π]
Try to get it in this Traditional form

5
π
8,0
G
7,8
-π
1,1,1,1,1,1,1
-5,0,0,0,0,0,0,0
​​
Where