Given the impulse train(Dirac comb):
$$\Delta_T(t)=\sum_{k\in\mathbb{Z}}\delta(t-kT)$$
where $T$ is the signal period, $\delta(t)$ is the Dirac delta function and $\mathbb{Z}$ is the set of integers
Now consider the Fourier series of $\Delta_T(t)$: $$ \Delta_T(t) = \frac{1}{T} \sum_{k\in\mathbb{Z}} e^{j2\pi\frac{k}{T}t} = \frac{1}{T} \left[ 1 + \sum_{k\in\mathbb{N}} \cos\left( 2\pi\frac{k}{T}t \right) \right] $$ Is there any way to prove that: $$\lim_{\epsilon\rightarrow 0} \int_{t-\epsilon}^{t+\epsilon}\Delta_T(\tau)d\tau= \begin{cases} 1 & \text{if } t=nT, \forall n\in\mathbb{Z}\\ 0 & \text{otherwise}\end{cases}$$ by passing through the Fouries series expansion?
In the figure is shown the previous integral for $\Delta_7(t)$ computed through the Fourier expansion truncated at th 15-th term
With suitable qualifications, of which there are several different possibilities, yes, your limiting "characterization" of the periodice Dirac delta can be understood from its Fourier series.
The auxiliary question, which in effect determines the terms in which the original question should be answered, is: what reasons or causal mechanisms would be considered acceptable to explain why the interchange of limits implied by (and which form of?) "passing through the Fourier series"?
If one does grant the legitimacy of computing those integrals via the Fourier series, I think summing geometric series gives the desired outcome.
If the question really is about justification of using the Fourier series to compute the integrals, which I suspect it is, then I'd tend to recommend thinking in terms of Sobolev spaces of functions on the circle... but/and this leads in a somewhat different direction...
(Perhaps the questioner can clarify...)