Relation between Fourier Series & Fourier transform
Regarding reciprocal domains, such as the "time" domain and "frequency" domain, that are related to each other by the Fourier Transform, whenever you uniformly sample in one domain, it causes periodic extension in the other domain.  It does that always.

So the Fourier Series is an example of sampling in the frequency domain causing periodicity in the time domain.  The corresponding example (using duality of the Fourier Transform) is the so-called DTFT (Discrete-Time Fourier Transform) $X(e^{j\omega})$ of a sampled sequence $x[n]$ which is naturally periodic with period $2 \pi$.  And normally we consider $X(e^{j\omega})$ with $-\pi &lt; \omega \le +\pi$ .