Fourier Transform of infinite sum I'm looking at last year's exams, and I found an exercise I can't solve:
(Roughly translated)

Consider $x(t)=\sum\limits_{k=1}^{+\infty}\left(\frac{1}{2}\right)^k \cos(k2\pi t)$ the input to a LTI system, with the transfer function shown in the picture below



a) Determine (and sketch) the Fourier transform of x(t) and y(t).

b) Determine the Fourier coefficients of y(t). Justify.



Edit:

The only thing that comes to mind is using the Linearity of the Fourier Transform and calculate the Transform of $\left(\frac{1}{2}\right)^k \cos(k2\pi t)$, k dependant, and $X(jw)=\sum\limits_{k=1}^{+\infty}X_k(jw)$, so it would be something like
$$
X(jw)=\sum\limits_{k=1}^{+\infty}\left[2^{-k}\pi\left[\delta(\omega-2k\pi) + \delta(\omega+2k\pi)\right]\right]
$$

However, I have never seen a Fourier Transform that looked like this, so I'm not quite sure that's correct, or, if it is, if it can be used (efficiently) for calculations, in this form.

After I get $X(jw)$, I can get $Y(jw)=X(jw)H(jw)$, so right now I'm pretty much just stuck in calculating $X(jw)$. Any help is much appreciated, and feel free to ask for clarification, if something I wrote is not clear enough.
i think it's pretty obvious that only the component with angular frequency of $4 \pi $ gets through the LTI system to the output.  so the output should be $\frac{1}{4} \cos(4 \pi t) $.