How to input MinRecursion to NIntegrate from NSum function

Question

Here is my source code:

λ=Exp[-9.966509]
s=1
r =0.101
η =0.01
k=1369
DConv[x_,λ_,s_,r_,η_,k_] := Exp[Log[Gamma[r+k]]-Log[k! Gamma[r]]-λ x -s Log[x] + r Log[r/(r+x η)] +k Log[x η/(r+x η)]]
NSum[DConv[x,λ,s,r,η,k],{x,1,Infinity}]/Re[PolyLog[s,Exp[-λ]]]

Bascially I want the he infinite sum of the DConv function but I am getting the following massage and the sum is also wrong:

NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option

I don't know how to set MinRecursion (which is an argument for NIntegrate) from NSum function.

Answer 1

Clear["Global`*"]

λ = Exp[-9.966509 // Rationalize[#, 0] &];
s = 1;
r = 101/1000;
η = 1/100;
k = 1369;

DConv[x_, λ_, s_, r_, η_, k_] =
  N[(Exp[Log[Gamma[r + k]] -
       Log[k! Gamma[r]] - λ x - s Log[x] +
       r Log[r/(r + x η)] +
       k Log[x η/(r + x η)]] //
     Simplify),
   30];

SetOptions[NIntegrate, MinRecursion -> 9];

sum = NSum[DConv[x, λ, s, r, η, k], {x, 1, Infinity},
    NSumTerms -> 50,
    WorkingPrecision -> 25]/Re[PolyLog[s, Exp[-λ]]] // N

(* 2.83045*10^-6 *)