Q: Does the series $\sum\limits_{k=1}^{\infty} \int\limits_0^1 \frac{\cos(2 k \pi x) \cos(2 n \pi x)}{k}\,\mathrm dx$ equal $0$ for $n \in \Bbb N$? Could someone confirm if my answer is correct? 
$$\sum_{k=1}^{\infty} \int_0^1 \frac{\cos(2 k \pi x) \cos(2 n \pi x)}{k}\,\mathrm dx=0, \quad n\in \mathbb{N}$$

A: $$I:=\int\limits_0^1\frac{\cos^22n\pi x}{n}dx$$
$$u:=2n\pi x\implies dx=\frac{du}{2n\pi}\implies$$
$$I=\frac1{2n^2\pi}\int\limits_0^{2n\pi}\cos^2u\,du=\left.\frac1{2n^2\pi}\left(\frac{u+\sin u\cos u}2\right)\right|_0^{2n\pi}=\frac1{2n^2\pi}\left(\frac{2n\pi}2\right)=\frac1{2n}$$
