I want to show that $\sum_n e^{ik n}$ is an infinite periodic sum of delta functions, where $n$ were integers from $-\infty$ to $\infty$.
I tried to manipulate the expression $\delta(x-a)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-a)dp}$ but without any luck.
To give you an idea of why it is true, recall Fourier series. Given a nice enough $2\pi$-periodic function $f : \mathbb{R} \to \mathbb{C}$ we can write $f(x) = \sum_{n \in \mathbb{Z}} c_n e^{inx}$ where $c_n = \frac{1}{2\pi} \int_{0}^{2\pi} f(x) \, e^{-inx} \, dx$. Now take $f(x) = \sum_{k \in \mathbb{Z}} 2\pi \, \delta(x-k)$, the $2\pi$-period extension of $\delta(x)$. Then $c_n = \frac{1}{2\pi} \int_{0}^{2\pi} 2\pi\,\delta(x) \, e^{-inx} \, dx = 1$ and we get $f(x) = \sum_{n \in \mathbb{Z}} e^{inx}.$