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I'm a beginner in the Fourier series and I can't find the solution to the below integral and relationship with the Fourier series. If

$$f(x)=1+\sum^\infty_{n=1}\frac{(-1)^n}{n}\cos(nx)+\frac{1}{n^2}\sin(nx)$$

calculate

$$I=\int^\pi_{-\pi}f(x)(\sin(2.5x)+\cos(2.5x))^2\cos(5x)dx.$$

Any help and hint are much appreciated.

Jessie
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hermi
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    Hint: Simplify out all all the trig in the integrand until you are left with $C_1\cos(ax)+ C_2\sin(bx)$. When you integrate a Fourier series against trig, it kills all the terms except for the terms in resonance aka the terms in the Fourier series that have the same $a$ and $b$. – Ninad Munshi Mar 07 '21 at 17:18
  • @NinadMunshi I simplify out all the trig and I achieve--> cos(5x) + 0.25*sin(10x). Doesn't this contradict the question? The Fourier series of the question has the same a and b. – hermi Mar 07 '21 at 17:48
  • Why would it contradict? If the Fourier series didn't have those $a$ and $b$, then the integral would be zero. – Ninad Munshi Mar 07 '21 at 18:09
  • @NinadMunshi Simplified phrase is cos(5x) + 0.25sin(10x) and question is pcos(nx) + q*sin(nx). Both resonance of the question are 'n' in the integral. – hermi Mar 07 '21 at 18:31
  • $n$ is not a variable that actually exists. What is the summation notation short for? Also btw the coefficient for $\sin 10 x$ is not correct – Ninad Munshi Mar 07 '21 at 18:31

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