Uniqueness of Fourier Series Representation and the Fourier Transform of Periodic Signals
The Fourier series, if it exists, is unique, apart from the fact that we can always use a multiple of the fundamental period $T$ as the period, or, equivalently, a fraction of the fundamental frequency $\omega_0$. The only consequence of this is that we introduce Fourier coefficients with value zero in between the non-zero coefficients. The values of the non-zero coefficients remain unchanged:

$$x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0}=\sum_{k=-\infty}^{\infty}\tilde{a}_ke^{jk\omega_0/N},\quad N\in\mathbb{Z}^+$$

$$\tilde{a}_k=\begin{cases}a_{k/N},\quad k=lN,\quad l\in\mathbb{Z}\\0,\quad\text{otherwise}\end{cases}$$