Dirac Distribution I'm trying to solve the following homework problem:

Show that,
  \begin{equation}
1 + 2\sum_{n=1}^{\infty} \cos(2\pi nx) = \sum_{k= - \infty}^{\infty}\delta(x-k),
\end{equation}
  in the sense of distribution.

I can show that the Dirac comb is given by the expression on the left by a Fourier series argument. However, I am not sure how to show this by a distribution argument. Any suggestions? Just give hints as this is a HW problem.
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$

$\ds{\sum_{k = -\infty}^{\infty}\delta\pars{x - k}}$ is
  even and periodic ( of period $\ds{1}$ ). Then,

\begin{align}
&\sum_{k = -\infty}^{\infty}\delta\pars{x - k} =
\sum_{n = 0}^{\infty}a_{n}\cos\pars{2\pi nx}
\\[1cm] &\
\int_{-1/2}^{1/2}\cos\pars{2\pi nx}\sum_{k = -\infty}^{\infty}\delta\pars{x - k}\dd x
\\[2mm] = &\
\sum_{m = 0}^{\infty}
a_{m}\underbrace{\int_{-1/2}^{1/2}\cos\pars{2\pi nx}\cos\pars{2\pi mx}\dd x}
_{\ds{=\ {1 + \delta_{n0} \over 2}\,\delta_{nm}}}
\\[5mm] &\
\underbrace{\int_{-1/2}^{1/2}\cos\pars{2\pi nx}\delta\pars{x}\dd x}_{\ds{=\ 1}} =
{1 + \delta_{n0} \over 2}\,a_{n}
\\[5mm] &\
\implies a_{n} = 2 - \delta_{n0}
\end{align}

$$
\implies
\bbx{\sum_{k = -\infty}^{\infty}\delta\pars{x - k} =
1 + 2\sum_{n = 1}^{\infty}\cos\pars{2\pi nx}}
$$
