Let $s:\Bbb R \to \Bbb R$ be the function $s(x)=\sum_{k=1}^\infty a_k \cos(kx)$ where $\sum_{k=1}^\infty a_k$ is an absolutely convergent series. Calculate $$\int_{-\pi}^\pi s(x) \ dx.$$
We want to compute $$\int_{-\pi}^\pi \sum_{k=1}^\infty a_k \cos(kx) \ dx$$ and what I would like to do is to conclude that $$\int_{-\pi}^\pi \sum_{k=1}^\infty a_k \cos(kx) \ dx= \sum_{k=1}^\infty a_k \int_{-\pi}^\pi \cos(kx) \ dx = \sum_{k=1}^\infty a_k \left(F(k\pi) - F(-\pi k) \right)$$ where $F$ is the antiderivative of $\cos$ i.e. $\sin$. So $$\int_{-\pi}^\pi s(x) \ dx = \sum_{k=1}^\infty a_k \left(\sin(k\pi) - \sin(-\pi k) \right) = 0.$$
Can I just use conclude this since of the absolute convergence here?
