Output of NIntegrate depends on MaxRecursion

I have an integral in this form:

NIntegrate[ (5184(-11 + k(98 + k(-16 + k(-40 + k(17 + 2k(7 + k)))))))/
((1 + E^((1/2)(-1 + k)))(1 + k)^3(-7 + k*(2 + k))^3), {k, 0, Infinity}]

when I set , MaxRecursion to 12 answer is of the order of 10^11 when set to 30 answer is of order 10^25 when set 40 order is 10^28 when set to 100 and above order is 10^34 and is constant up to 400 but I have an error in answer.

How should I proceed to get the correct result?

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– rhermans
Nov 26 ’14 at 12:00

  

 

You get “an error” that actually contains information “Integrate failed to converge to prescribed accuracy after 100 recursive bisections”.
– rhermans
Nov 26 ’14 at 12:14

  

 

yes rhermans this is my integral error.
– Ehsan F
Nov 26 ’14 at 12:16

1

 

The function diverges at (22–√−1)(2 \sqrt{2}-1), and it looks like the integral also diverges, the answer then is Infinity
– rhermans
Nov 26 ’14 at 12:29

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2 Answers
2

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There are too many singularities in your integrand (according to Reduce there are infinity of them, but these are complex valued) and then there are 3 real valued poles {-1., -3.82843, 1.82843} (I did not check for zero/pole cancellations). So the only real pole in the range of the integration is 1.82843 or -1 + 2 Sqrt[2]

integrand = 5184 ((-11 + k (98 + k (-16 + k (-40 + k (17 + 2 k (7 + k))))))/
((1 + E^(1/2 (-1 + k))) (1 + k)^3 (-7 + k (2 + k))^3));
poles = Reduce[Denominator[integrand] == 0, k]

So trying to integrate this is a losing battle. But you can get little better numbers if you exclude the -1 + 2 Sqrt[2] and increase WorkingPrecision. At least the order of the numerical result does not change as much.

NIntegrate[integrand, {k, 0, Infinity}, Method -> PrincipalValue,
Exclusions -> -1 + 2 Sqrt[2], MaxRecursion -> 12, WorkingPrecision -> 100]

With MaxRecursion -> 30

With MaxRecursion -> 40

With MaxRecursion -> 100

btw, Integrate says also it does not converge:

Integrate[integrand, {k, 0, Infinity}]

So may be you need to change the integrand?

The consistent behavior displayed by the OP’s integral — increase recursion, increase the magnitude of the integral — normally is the result of a divergent integral. It’s possible the OP seeks the principal value of the integral. There is one pole of order 3 in the interval of integration

f = (5184 (-11 + k (98 + k (-16 + k (-40 + k (17 + 2 k (7 + k))))))) /
((1 + E^((1/2) (-1 + k))) (1 + k)^3 (-7 + k*(2 + k))^3);

Solve[Denominator[f] == 0 && k > 0, k, Reals]
pole = k /. First[%];

(* {{k -> -1 + 2 Sqrt[2]}, {k -> -1 + 2 Sqrt[2]}, {k -> -1 + 2 Sqrt[2]}} *)

Since the order is odd there is some hope that the principal value exists. But it does not exist since the series expansion shows that the term of degree -2 does not vanish:

N@Series[f, {k, pole, 0}]

If one is interested in further confirmation, subtract out the divergent term and see whether the integral converges:

coeff = SeriesCoefficient[f, {k, pole, -2}];
NIntegrate[f – coeff/(k – pole)^2, {k, 0, pole, Infinity}, Method -> “PrincipalValue”]

(* -277.122 *)