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

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.

• First, you have mismatched parenthesis somewhere. Second, your function [Nu][n] needs a definition. Jul 8 '16 at 15:36
• I Forgot, [Nu][n_] := (2*n*[Pi]) And the parentesis is of [omega] that i mismatched. But the problem persist Jul 8 '16 at 15:58
• Is Beta defined? Jul 8 '16 at 16:15
• Yes, Sorry, I Choose [Beta]=1 Jul 8 '16 at 16:25
• with V=1/4; F1[n_]:=NSum[Simplify[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}]; F1 it quickly concludes ComplexInfinity
– Bill
Jul 8 '16 at 17:02

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.

• Now everything is working. In fact there is a problem with derivatives on this function. I'm in debit with you Jul 8 '16 at 18:28