Why is the continuous time Fourier series of DC signal an impulse?
CTFS , $a_k$, of a periodic function $x(t)$ of period $T$ is :
$$ a_k = \frac{1}{T} \int_{&lt;T&gt;} x(t) e^{-j \frac{2\pi}{T} k t } dt \tag{1} $$
and the DC coefficient corresponds to setting $k=0$ above;
$$ a_0 = \frac{1}{T} \int_{&lt;T&gt;} x(t) dt \tag{2} $$
where the integration is along one period of $x(t)$. The DC coefficient cannot be an impulse, (indeed the CTFS coefficients $a_k$ form a discrete sequence, in $k$, of finite amplitude by definition of its existance.)
A signal must have finite energy per period (aka finite avg power) in order to have a valid CTFS (due to Dirichlet conditions) representation, hence the above integral in Eq.2 should be finite; i.e., the integral converges, and cannot result in an impulse.