# Frequency multiples in Fourier Series expansions

When we expand signal with Fourier series, why angular frequencies are in multiples? I.e. why $$x(t)=\sum_{k=-\infty}^{\infty}z_k*e^{jk\omega_0t}=z_0+z_1e^{j\omega_0t}+z_2e^{j2\omega_0t}+z_3e^{j3\omega_0t}+...$$ (negative terms are left out in order to save space).

My question is why frequencies are multiples of each other. For example why don't we expand in the following way: $$x(t)=z_0+z_1e^{j\omega_0t}+z_2e^{j2.4\omega_0t}+...$$ (note that we used $$j2.4\omega_0$$ instead of $$j2\omega_0$$)

In fact, the coefficients of the Fourier Series can be seen as the Fourier Transform of the periodic signal evaluated over a single period, for integer multiples of the fundamental frequency. Whereas the Fourier Transform of a signal is an integral evaluated over all $$\mathbb{R}$$, with omega also taking any value in $$\mathbb{R}$$.