Find the Fourier Series of the function $p(x)=\frac{\cos(5\pi x)+e^{-i\pi x}+1}{2}$ and justify why $p(x)$ must equal its Fourier series.
This is a review problem for my Analysis exam.
My idea was to compute the Fourier coefficients $c_n$, and show that the sum $\sum_{n=-\infty}^{\infty} |c_n|$ is absolutely convergent, so that the Weierstrass M-test can be applied to demonstrate uniform convergence of the Fourier series (and thus, pointwise convergence).
However, I am struggling to compute the coefficients and find a suitable sequence of 'bounding' terms.
I tried: $$c_n=1/2\int_{[0, 1]}(e^{5 i \pi x -2 \pi i n x}+e^{-5 i \pi x-2 \pi i n x})/2+e^{-i\pi x - 2 \pi i n x}+e^{-2\pi i n x} dx$$
When $n=0$, I got $c_0=1/(i \pi)+1/2$. When $n \neq 0$, I got $c_n=\frac{2 n i}{\pi(25-4n^2)}-\frac{i}{2\pi n + \pi}.$
I'm stuck trying to show that the series of $|c_n|$ converges absolutely, and I'm questioning if this approach the correct one, since the sequence of $|c_n|$ looks like it grows similarly to the harmonic series.
Note: My textbook defines $c_n=\int_{[0, 1]}f(x)e^{-2\pi i n x}$ dx