How to prove this equality about series with exponential function?

Question

I need to prove this $$\frac{1}{N+1}\left| \sum \limits_{k=0}^N e^{2\pi i m k x}\right|^2 =\sum \limits_{n=-N}^{N}\left( 1-\frac{|n|}{N+1}\right)e^{2\pi i m n x}$$

Answer 1

Hint: Using obvious notation, $$ \left|\sum_k a_k\right|^2=\left(\sum_k a_k\right)\overline{\left(\sum_j a_j\right)}=(\sum_k a_k)(\sum_j\overline{a_j})=\sum_k\sum_j a_k\overline{a_j}. $$ The sums over $k$ and $j$ each run from $0$ to $N$, so that $k-j$ takes integer values from $-N$ to $N$. Group by $k-j$: $$ \sum_k\sum_j a_k\overline{a_j}=\sum_{n=-N}^N\sum_{k-j=n}a_k\overline{a_j}, $$ where the inner sum $\sum_{k-j=n}$ means "sum over all $k$, $j$ pairs such that $k-j=n$". How many $k,j$ pairs have this property, and what is the value of $a_k\overline{a_j}$ for each of these pairs?