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I tried to compute

$$\int_0^{2\pi}\left(\sum_{k=0}^n\cos(k x)\right)\mathrm dx$$

with two different codes:

Integrate[Sum[Cos[k*x],{k,0,n}],{x,0,2*Pi},Assumptions->n\[Element]Integers && n>0]

and

Refine[Integrate[Sum[Cos[k*x],{k,0,n}],{x,0,2*Pi}],n\[Element]Integers && n>0]

but both failed. However if I make a table of values for n then the computation works normally. There is a way to fix this behavior?

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The integral is so simple and the interchangeability of summation and integration is so obvious to a human but not always to a computer, especially with an undefined symbolic number of summation terms. I hope in future this will just work, but for you to get results try making tiny simple changes that simplify things dramatically.

Sum[Integrate[Cos[k*x], {x, 0, 2*Pi}], {k, 0, n}]

$2 \pi$

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  • $\begingroup$ Yes, it is obvious in this case... but I want to know if I can fix this behavior for other cases. $\endgroup$ – Masacroso Apr 3 '17 at 10:45
  • $\begingroup$ @Masacroso Why if interchangeability of summation and integration always works? $\endgroup$ – Vitaliy Kaurov Apr 3 '17 at 10:46
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    $\begingroup$ I dont knew that changing the order it can fix this behavior, thank you. Probably what happens is that mathematica dont know that k is a natural number for the other computation order. $\endgroup$ – Masacroso Apr 3 '17 at 10:47

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