Prove $\int_{-\pi}^\pi F_n(y) \, dy=1$

Hint: Let $$D_n(x)= \sum_{k=-n}^n e^{ikx},$$ and let $$F_N(x) = \sum_{n=0}^{N-1} D_n(x).$$ Prove that $$ F_N(x) = \frac{1}{N}\frac{\sin^2 (Nx/2)}{\sin^2 (x/2)}.$$

$D_n$ is known as the Dirichlet kernel and $F_N$ is known as the Fejér Kernel.

You have

$$F_n(x) = \frac{2\pi}{n+1}D_n^2(x)=\frac{1}{2\pi (n+1)}\left(\sum_{k=-n}^n e^{ikx} \right)^2= \frac{1}{2\pi (n+1)}\frac{\sin^2 \left(\frac{(n+1)x}{2}\right)}{\sin^2 \left(\frac{x}{2}\right)}.$$

from which follows by switching $\int$ and $\sum$ the equalities

$$\begin{aligned}\int_{-\pi}^{\pi}F_n(x) \ dx&= \frac{1}{2\pi (n+1)}\int_{-\pi}^{\pi}\left(\sum_{k=-n}^n e^{ikx}\right)^2 \ dx\\ &= \frac{1}{2\pi (n+1)}\int_{-\pi}^{\pi}\sum_{k=-n}^n \sum_{l=-n}^ne^{i(k+l)x}\ dx\\ &= \frac{1}{2\pi (n+1)}\sum_{k=-n}^n \sum_{l=-n}^n \int_{-\pi}^{\pi}e^{i(k+l)x}\ dx\\ &=1 \end{aligned}$$

as in the double sum, the only non vanishing terms are for $k=-l$ and there are $n+1$ such terms.