In[]:=
M2-
25
∑
n=1
N
1-n

π
f[n],45
In[]:=
M2=NIntegrate[E^(IPix)(x^(1/x)-1),{x,1,InfinityI},WorkingPrecision->100];
In[]:=
M2
Out[]=
0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821-0.04738061707035078610720940650260367857315289969317363933196100090256586758807049779050462314770913485
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,14}];​​N[M2-c,60]]
Out[]=
{114.141,-2.68061752061682413451496324777357274893572822802243467155680×
-20
10
+5.46343531980465889484729016851138946513767493900692074798878×
-20
10
}
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=FunctionExpand[Sum[(I/π)^(1-x)f[x],{x,1,14}]];​​N[M2-c,30]]
N
:Internal precision limit $MaxExtraPrecision = 50. reached while evaluating
2034715951
6
π
(4299761999190945178422296607447983849472000000000+13+1421878967986423670113193322568777728000000000
3
EulerGamma
3
π
+194)
76564527432102820851321962773552640777256960000000000000000000
.
N
:Internal precision limit $MaxExtraPrecision = 50. reached while evaluating --
π
3
+
2
π
4
-
EulerGamma
2
π
6
+
139
3
π
648
-
11
108
EulerGamma
3
π
+
1
36

2
EulerGamma
3
π
-
12181
4
π
124416
+
703EulerGamma
4
π
10368
-
25
2
EulerGamma
4
π
1728
+
3
EulerGamma
4
π
432
+1257Zeta[3].
N
:Internal precision limit $MaxExtraPrecision = 50. reached while evaluating -

3
π
30
-
5
4
π
288
+
EulerGamma
4
π
120
-
21019
5
π
2160000
+
137EulerGamma
5
π
36000
-

2
EulerGamma
5
π
1200
+
400607
6
π
129600000
-
24289EulerGamma
6
π
12960000
+
49
2
EulerGamma
6
π
144000
-
3
EulerGamma
6
π
21600
+584Zeta[5].
General
:Further output of N::meprec will be suppressed during this calculation.
Out[]=
{164.594,-2.68061752061682413451496324777×
-20
10
+5.46343531980465889484729016851×
-20
10
}
​​​​
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,15}];​​N[M2-c,60]]
Out[]=
{206.828,-7.0048557433051403089887845374418782449539593967029838001456×
-22
10
+1.26736626874507997746603702354372817414155093401053285040001×
-21
10
}
​​​​
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,16}];​​N[M2-c,60]]
Out[]=
{367.656,-1.68741401983354402198637018860025140436294362391490862102914×
-23
10
+2.76362031569291751104269409539867188424537461354264473659699×
-23
10
}
​​​​
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,17}];​​N[M2-c,60]]
Out[]=
{648.922,-3.77933455462791805456776366309668257576294430136732731280126×
-25
10
+5.68664402387510462101950493523535652853536708267537177946343×
-25
10
}
​​​​
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,18}];​​N[M2-c,60]]
Out[]=
{1144.06,-7.92162979800545585897916906818537276469654387317051030478407×
-27
10
+1.107884180082112247993943979438617775110956621372671217532984×
-26
10
}
​​​​
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,19}];​​N[M2-c,120]]
N
:Internal precision limit $MaxExtraPrecision = 50. reached while evaluating (0.01315113632289691589604247959037919771223845025998154152767816536478480212499350597973208796449221096 - 0) + <<18>>.
Out[]=
{2053.44,-1.5619894076106487865898134025824912807464108715702669736910305072986383190×
-28
10
+2.0496756136125713793076747790675356922545792073520786233859343700968744330×
-28
10
}
​​​​
In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,20}];​​N[M2-c,240]]
N
:Internal precision limit $MaxExtraPrecision = 50. reached while evaluating (0.01315113632289691589604247959037919771223845025998154152767816536478480212499350597973208796449221096 - 0) + <<19>>.
Out[]=
{3684.34,-2.90981083913879440719163112473792641560032218159284674054623668976206556×
-30
10
+3.61060541537316437493018595910109302892451592999320615986906112823294257×
-30
10
}
​​​​
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,21}];​​N[M2-c,240]]
In[2]:=f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-Iπ];​​Timing[c=Sum[(I/π)^(1-x)FunctionExpand[f[x]],{x,1,21}];​​N[M2-c,240]]​​​​Out[3]={6735.36,(-0.0707760393115288035395280218303333988607106741750348325523678799979663251738938098297290694459097184284355027619836454922180137012622310989124546482243158293359255800744150590003345926169342157982495044263119906516990516625263234761771674761+0.0473806170703507861072094065026643818648796664752316411010652824733741785739453736701429145725055707230016771207995975088908282595699190561632209174081864687767531530082929114554819802540324680460121670751536237698239718588779330762311416834I)+M2}​​​​In[10]:=Last[%3]​​​​Out[10]=-5.139749495597797167180496919509183434786790951177324889718817973248022*10^-32+6.070329172676678205800176910428157080831098587487587963829142479643588*10^-32I
This is checked very easily and " infinitely" faster below.
In[]:=
Timing[​​​​M2-(NIntegrate[​​​​Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,21}]],{x,​​​​1,InfinityI},WorkingPrecision->40])]​​​​​​
Out[]=
{0.265625,-5.139749496*10^-32+6.070329173*10^-32I}