NIntegrate inside NSum

Question

Consider the following function with a numerical integration:

BDMAF[n_, \[Gamma]_, x_, c_] := 
 Module[{K1 = EllipticK[1/Sqrt[1 + \[Gamma]^2]], 
   E1 = EllipticE[1/Sqrt[1 + \[Gamma]^2]], y0},
  y0 = ((\[Pi]^2 (IntegerPart[x + 1] - x + n))/(2 K1 E1))^(1/2);
  Exp[-y0^2] NIntegrate[
    Exp[y^2] Sin[(n + IntegerPart[x + 1])/2 \[Pi] + c y]^2, {y, 0, y0}]
  ]

that works perfectly if all parameters are explicitly defined:

BDMAF[1, 1, 1, 1]

0.0391914

I need to calculate the infinite sum over n. Naive approach gives

NSum[BDMAF[n, 1, 1, 1], {n, 0, \[Infinity]}]

NIntegrate::nlim: y = 1.38268 Sqrt[1. +n] is not a valid limit of integration. >>

Following the answer to the question about nested NIntegrate, I tried to redefine BDMAF as follows:

BDMAF[n_?NumericQ, \[Gamma]_, x_, c_] := BDMAF[n, \[Gamma], x, c] = 
  Module[{K1 = EllipticK[1/Sqrt[1 + \[Gamma]^2]], 
    E1 = EllipticE[1/Sqrt[1 + \[Gamma]^2]], y0},
   y0 = ((\[Pi]^2 (IntegerPart[x + 1] - x + n))/(2 K1 E1))^(1/2);
   Exp[-y0^2] NIntegrate[
     Exp[y^2] Sin[(n + IntegerPart[x + 1])/2 \[Pi] + c y]^2, {y, 0, y0}]
   ]

The problem remains, but I receive another error message:

NSum[BDMAF[n, 1, 1, 1], {n, 0, \[Infinity]}]

NIntegrate::inumr: The integrand BDMAF(n,1,1,1) has evaluated to non-numerical values for all sampling points in the region with boundaries (15. 4.64782*10^14). >>

How BDMAF function should be defined?

Answer 1

Problem was in the non-convergent sum. This example with the new sum works good:

BDMAF[n_?NumericQ, \[Gamma]_, x_, c_] := BDMAF[n, \[Gamma], x, c] = 
  Module[{K1 = EllipticK[1/Sqrt[1 + \[Gamma]^2]], 
    E1 = EllipticE[1/Sqrt[1 + \[Gamma]^2]], y0},
   y0 = ((\[Pi]^2 (IntegerPart[x + 1] - x + n))/(2 K1 E1))^(1/2);
   Exp[-y0^2] NIntegrate[
     Exp[y^2] Sin[(n + IntegerPart[x + 1])/2 \[Pi] + c y]^2, {y, 0, y0}]
   ]

NSum[Exp[-Pi n] BDMAF[n, 1, 1, 1], {n, 0, Infinity}]

Thanks for your attention.