We typically use the Fourier Series (FS) to represent periodic signals. When we do, the Fourier Series representation is equal to the signal for all time.
For non-periodic signals, we can still use Fourier Series to represent the signal on some time interval. The time interval we choose will set the periodicity of the FS representation. Outside of our expansion interval the original signal and the FS will not equal each other, but they will on the expansion interval.
This video works a specific example of finding the FS representation of the continuous-time signal x(t) = exp(-alpha*t) on the time interval 0 to 10. After computing the Fourier Series Coefficients, we plot the FS representation for different numbers of terms in the summation to see how the representation converges to the desired signal.