Error "Summand (or its derivative)...is not numerical at point m = -85." on a simple sum

Question

I'm a beginner on mathematica.. I'm trying to calculate a simple sum of a function,

F1[n_]:=NSum[(V^2*8)/d^2*(I*ω[m]-I*Sign[ω[m]]*Sqrt[(ω[m])^2 +
  (d/2)^2])*1/(I*ω[m]+I*ν[n]-Σf), {m, -Infinity, Infinity}]

Where:

ω[n_] := ((2*n + 1)*π)/β
ν[n_] := (2*n*π)/β
d=12
V=0.25
Σf = 1 + I
β=1

For n=2, for example. But when I try to do this it returns an error message:

"Summand (or its derivative)...is not numerical at point m = 15 "

But when I choose upper and lower bound of simulation with the function:

F2[n_, min_, max_] :=NSum[(V^2*8)/d^2*(I*ω[m] - 
  I*Sign[ω[m]]*Sqrt[(ω[m])^2 + (d/2)^2])*1/(I*ω[m] + I*ν[n] - Σf), {m, min, max}]

Mathematica returns a value for small limits, for example:

F2[2, -10, 10] = 0.000757504 - 0.00143062 I

But, for: F2[2, -100, 100] I have a similar error mensage.

"Summand (or its derivative)...is not numerical at point m = -85."

I will be grateful if someone help-me on this calculations of infinity sum.

Answer 1

Try changing the method of summation:

Clear["Global`*"]

β = 1;
d = 12;
V = 1/4;
Σf = 1 + I;
ω[m_] := ((2*m + 1)*π)
ν[n_] := (2*n*π)/β

F1[n_] := 
 NSum[(V^2*8)/
    d^2*(I*ω[m] - 
     I*Sign[ω[m]]*
      Sqrt[(ω[m])^2 + (d/2)^2])*1/(I*ω[m] + 
      I*ν[n] - Σf), {m, -Infinity, Infinity}, 
  Method -> "WynnEpsilon", WorkingPrecision -> 30]

0.0004521 - 0.0014314 I

Or increasing NSumTerms:

F2[n_, min_, max_] := 
 NSum[(V^2*8)/
    d^2*(I*ω[m] - 
     I*Sign[ω[m]]*
      Sqrt[(ω[m])^2 + (d/2)^2])*1/(I*ω[m] + 
      I*ν[n] - Σf), {m, min, max}, 
  NSumTerms -> 200000]

F2[2, -100000, 100000]

0.000450002 - 0.00143142 I

This post was helpful: Is this a bug of NSum?

Since NSum[] uses Euler-Maclaurin summation as its default, and since that method involves taking the derivative of the summand, you get what you observe. -J.M.