signal analysis - What is difference between outputs of Fourier transform and Fourier series of a periodic square waveform?

We can use Fourier transform of an aperiodic signal and Fourier series of periodic signal. But we can use Fourier transform formula for periodic function also.

Now, let us consider a periodic square wave with fundamental period $T$ . Then I want to ask is

What is difference between the outputs of Fourier transform and Fourier series of a periodic square waveform?

Answer

The Fourier transform $X(\omega)$ of a $T$-periodic function $x(t)$

$$ x(t+T) = x(t) \quad \quad \forall t $$

having complex Fourier coefficients $c_n$

$$ x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j 2 \pi n t/T} $$

$$c_n = \frac{1}{T} \int_{0}^T x(t) e^{-j 2 \pi n t/T} \ dt \tag{1}$$

can be expressed as a weighted sum of Dirac impulses, where the weights are given by the complex Fourier coefficients:

$$X(\omega) \triangleq \mathcal{F}\left\{ x(t) \right\} = 2 \pi \sum_{n=-\infty}^{\infty} c_n \delta\left(\omega - \frac{2\pi n}{T}\right) \tag{2}$$