Fourier Transform of a Periodic function

Question

I have some questions regarding Fourier series and the Fourier transform.

The frequencyspectum consists of the Cn's of the Fourier series of a periodic function. The Fourier transform does the same thing for non-periodic functions.

Now does the Fourier transform of a PERIODIC function give the same spectrum as those Cn terms ?

Extra question: we should be able to draw the spectrum in our classes. With the frequency on the x-axis and |Cn| on the y-axis. If Cn is complex or negative does that mean I still draw a line upwards because of |Cn| ?

Thank you all

Answer 1

Here is a Fourier spectrum of a periodic function $f(x) = \cos(kx+1)^{24}$, in theory we have infinitesmally tight spikes at even intervals and 0 everywhere else. If you have learned distributions those are called Dirac distributions but numerically often one will have to be satisfied with a very pointy peak in the spectrum of the discrete fourier transform because the spectrum must align perfectly to fit theory.

As you can see there is the same distance in frequencies between each peak. That is what happens when doing the Fourier Transform to a function which has a fourier series.