I feel like this is a question with an easy answer but I cannot find an explanation. In the connection between the Fourier Series and the Fourier transform, it is shown that any periodic function is the convolution of a dirac comb (impulse train) with the function that is desired. However, in order to show the results, you need to know the Fourier Transform of the Dirac Comb, but every derivation I found calculates the fourier series first. I tried to run the calculation myself but I get stuck.
Here is my attempted solution:
$\mathcal{F}(\sum_{n=-\infty}^{n = \infty} \delta(t-nT)) =$
$\sum_i \int_{-\infty}^{\infty} \delta(t-nT)e^{-i2\pi f' t} dt=$
$\sum \int\int e^{2\pi i f(t-nT)}e^{-i2\pi f' t} dt df $
$= \sum \int \delta(f-f')e^{-i2\pi nTf'}df = \sum e^{-i2\pi f' nT}$
What am I doing wrong? Is this even valid?