θ=(7/8);ν=(3/4);

{θ,ν}//N

{(θ/(1-θ)),(ν/(1-ν))}

ξ=θ/ν

N[%]

Remove[P]

P[n_]:=P[n]=(1+n) (((-1+ν))^2) (ν^n)

P[n_]:=(1-θ)(1-ν)(((θ^(n+1))-(ν^(n+1)))/(θ-ν))

NumberForm[N[lξ],{4,3}]

ToString[NumberForm[N[ξ],{4,3}]]

nmax=20;

LogGrid[lowerbound_,upperbound_,numpts_,accuracy_]:=Module[{grid,lb=Log[lowerbound],ub=Log[upperbound]},grid=Rationalize[#,0]&/@Table[Exp[mPb],{mPb,lb,ub,(ub-lb)/(numpts-1)}];
If[accuracy!=0,Round[grid,accuracy],grid]]

ClearAll[translations]
translations[levels:{__}:{1},dir:"X"|"Y"|"Z":"X",h_:(0&)]:=ReplaceAll[Point[x_]:>Module[{d=dir/. Thread[{"X","Y","Z"}->Range[3]]},{Point@#,Line@#}&/@Table[Insert[#,i+h@@#,d]&/@x,{i,levels}]]];

kmax=50; (*sum for convergence*)

PlotFrameStandardized[ξ_]:=Module[{dm,lpa,pa,lpb,pb,pr=.07(*plotrange*),lc={83,37,127}/255(*color*)},
dm=ListPlot3D[Table[((ξ-1)/((ξ^(n+m+1))-1))(ξ^n)  P[n+m],{n,0,nmax},{m,0,nmax}],
InterpolationOrder->0,Mesh->None,Filling->Axis,FillingStyle->Opacity[.2],PlotRange->All,ViewPoint->{1.7669096015509167`,2.027912904012787`,2.053192566148043`},ViewVertical->{-0.3986024860842796`,-0.4574830112374552`,0.7948744256274334`},PlotStyle->Lighter[RGBColor[lc[[1]],lc[[2]],lc[[3]]]],Background->White,Epilog->Inset["ξ="<>ToString[NumberForm[N[ξ],{3,2}]],{1,0}],AspectRatio->1];
lpa=ListPlot[Table[{n,pr(((ξ^n) (∑_(k=n)^kmax)((ξ-1)/((ξ^(k+1))-1))P[k])/(Max[Table[(ξ^n) (∑_(k=n)^kmax)((ξ-1)/((ξ^(k+1))-1))P[k],{n,0,nmax}]]))},{n,0,nmax}],PlotStyle->{Red,Thickness[.0075]}];
pa=Graphics3D[translations[{nmax+1}]@lpa[[1]],BoxRatios->{2,1,1},ImageSize->Large,Axes->True];
lpb=ListPlot[Table[{n,pr(((ξ^(-n)) (∑_(k=n)^kmax)((ξ-1)/((ξ^(k+1))-1))(ξ^k)P[k])/((ξ^(-0)) (∑_(k=0)^kmax)((ξ-1)/((ξ^(k+1))-1))(ξ^k)P[k]))},{n,0,nmax}],PlotStyle->{Blue,Thickness[.0075]}];
pb=Graphics3D[translations[{nmax+1},"Y"]@lpb[[1]],BoxRatios->{2,1,1},Axes->True];
ImagePad[Show[dm,pa,pb,PlotRange->{0,pr},BaseStyle->36,ImageSize->1200,ImageResolution->1200],200,White]
]

PlotFrameStandardized[1.2345]

Export["frame"<>ToString[25]<>".png",PlotFrameStandardized[lξ[[25]]],"PNG",ImageSize->900]

θ/ν//N

NotebookDirectory[]

lξup=LogGrid[θ/ν,100,50,(10^(-5))]//N

lξ=Delete[Join[lξup,Reverse@lξup],-1]

ξmax=25;

With a sigmoid easing ramp:

lξup=Table[(θ/ν)+ξmax(((Tanh[x]+1))/1),{x,-5,2,(5-(-2))/25.}]

lξ=Prepend[Delete[Delete[Join[lξup,Reverse@lξup],-1],1],(θ/ν)]

ListPlot[lξ]

It’s tricky to have a nice result because the response itself is not linear in log-scale on both limits, so one has to tune by hand the amount of time spent in each case:

ListPlot[Join[lξ,lξ,lξ,lξ]]

NotebookDirectory[]

Directory[]

Do[Export["frame"<>ToString[iξ]<>".png",PlotFrameStandardized[lξ[[iξ]]],"PNG",ImageSize->900],{iξ,Length@lξ}]