In[]:=
mm=NSum[
n
(-1)
(
1/n
n
-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->200];
In[]:=
Print[Style["in  the denominator of f, Sin[Pi u] looks similar to -Re[​-1
u-Pi/2
)
​]:",FontSize->24]];​​(*Definethefunctionsbandc*)b[x_,k_]:=x^2+1-c[x,k];​​c[x_,k_]:=(((k)^2x^2-1)(x-1)^2)/((2k+1)x(x+1)-If[k<100,c[x,k+1],0]);​​​​(*Definef1[x_]usingTable*)​​f1[x_]:=2(x-1)/b[x,1]​​(*DefineActualvaluesusingTable*)​​
in  the denominator of f, Sin[Pi u] looks similar to -Re​-1
u-Pi/2
)
​:
In[]:=
Im[(-1)^(u)/I]
Out[]=
-Re[
u
(-1)
]
In[]:=
Plot[Sin[Piu],-Re[
u-Pi/2
(-1)
],{u,0,1}]
In[]:=
ReImPlot[Sin[Piu],-Re[
u-Pi/2
(-1)
],{u,0,1/1000}]
Out[]=
In[]:=
Plot[Sin[Piu],-Re[
u-Pi/2
(-1)
],{u,0,40}]
Out[]=
In[]:=
Plot[Sin[Piu],-Re[
u-Pi/2
(-1)
],{u,40,41}]
Out[]=
​​​​​​​​​​
​​​​
In[]:=
Print[Style["Using the Burns, Mathematica, AI, Continued Fraction method",FontSize->24]];​​(*Definethefunctionsbandc*)b[x_,k_]:=x^2+1-c[x,k];​​c[x_,k_]:=(((k)^2x^2-1)(x-1)^2)/((2k+1)x(x+1)-If[k<400,c[x,k+1],0]);​​​​(*Definef1[x_]usingTable*)​​f1[x_]:=2(x-1)/b[x,1]​​(*DefineActualvaluesusingTable*)​​
Using the Burns, Mathematica, AI, Continued Fraction method
In[]:=
Timingmm-ImNIntegrate
f1[u+1]
Sin[Piu]
,{u,0,100(I)},WorkingPrecision->200
Out[]=
{39.25,-1.080747246452699010782561061749520918354869201770563381728258×
-138
10
}
In[]:=
Timingmm-ImNIntegrate
f1[u+1]
Sin[Piu]
,{u,0,120(I)},WorkingPrecision->200
Out[]=
{61.9063,-3.7168370924733104155677743436475441929604296335101754431×
-143
10
}
In[]:=
Timingmm-ImNIntegrate
f1[u+1]
Sin[Piu]
,{u,0,129(I)},WorkingPrecision->100
Out[]=
{29.7969,2.91025632077345248908118083416452562803548052068571863822408704449163947255008993×
-20
10
}
In[]:=
mm=NSum[
n
(-1)
(
1/n
n
-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->2000];Print[Style["k<600",FontSize->24]];​​(*Definethefunctionsbandc*)b[x_,k_]:=x^2+1-c[x,k];​​c[x_,k_]:=(((k)^2x^2-1)(x-1)^2)/((2k+1)x(x+1)-
If[k<600,c[x,k+1],0]
)
;​​​​(*Definef1[x_]usingTable*)​​f1[x_]:=2(x-1)/b[x,1]​​(*DefineActualvaluesusingTable*)​​
k<600
In[]:=
Timingmm-ImNIntegrate
f1[u+1]
Sin[Piu]
,{u,0,129(I)},WorkingPrecision->100
In[]:=
mm-TableImNIntegrate
f1[u+1]
Sin[Piu]
,{u,0,n(I)},WorkingPrecision->500,{n,128,131}//Quiet;
In[]:=
p=N[%,20]
Out[]=
{-6.5852344354704877656×
-270
10
,-2.8339518551133830063×
-271
10
,-1.2196142233666113792×
-272
10
,-5.2488163150454726828×
-274
10
,-2.2589847751228051817×
-275
10
}
In[]:=
mm=NSum[
n
(-1)
(
1/n
n
-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->2000];Print[Style["k<700",FontSize->24]];​​(*Definethefunctionsbandc*)b[x_,k_]:=x^2+1-c[x,k];​​c[x_,k_]:=(((k)^2x^2-1)(x-1)^2)/((2k+1)x(x+1)-
If[k<700,c[x,k+1],0]
)
;​​​​(*Definef1[x_]usingTable*)​​f1[x_]:=2(x-1)/b[x,1]​​(*DefineActualvaluesusingTable*)​​
k<700
In[]:=
mm-TableImNIntegrate
f1[u+1]
Sin[Piu]
,{u,0,n(I)},WorkingPrecision->500,{n,132,146}//Quiet;
In[]:=
N[p=%,20]
Out[]=
{-1.8947253250036587337×
-182
10
,-8.1384160414959303392×
-184
10
,-3.4958464643138042134×
-185
10
,-1.5017006785488801210×
-186
10
,-6.4510840694989621363×
-188
10
,-2.7714054336692588606×
-189
10
,-1.1906529986926414492×
-190
10
,-5.1154976604209067167×
-192
10
,-2.1978996562069046247×
-193
10
,-9.4437587353323187241×
-195
10
,-4.0578755482593537300×
-196
10
,-1.7436894990992913483×
-197
10
,-7.4930036609353832368×
-199
10
,-3.2200222642146523021×
-200
10
,-1.3838135005261082632×
-201
10
}