The Context:
Find $X(ω)$ which is the frequency domain representations of $x(t)$.
$$ x(t) = \sum_{k=-\infty}^\infty \delta(t-4k) $$
This my professor's solution:
As we can see, the way he proceeded is as follows: 1) Since x(t) is periodic, he first calculated the Fourier Series coefficient of x(t). 2) He used the following formula which allows to convert from Fourier Series to Fourier Transform:
a_k represents the Fourier series coefficient.
My Question:
Why can't this same result be obtained using the Fourier Transform Analysis equation or the Fourier Transform Tables?
For example, using the Fourier Transform Analysis Equation, we obtain the following.
As we can see, the result obtained does not seem to be the same as my professor's. Or is it? Are the two results equivalent if we were to somehow simplify my answer algebraically? If they're not equivalent, how would I know which method to use, mine or my professor's, to solve this type of problem? Do I use my professor's method because x(t) is periodic?