You correctly figured out that the occurring integrals don't converge in the conventional sense. The easiest (and definitely non-rigorous) way to see the result is by noting the Fourier transform relation
$$1\Longleftrightarrow 2\pi\delta(\Omega)$$
By the shifting/modulation property we have
$$e^{j\Omega_0t}\Longleftrightarrow 2\pi\delta(\Omega-\Omega_0)$$
So each term $e^{jn\Omega_s t}$ in the Fourier series transforms to $2\pi\delta(\Omega-n\Omega_s)$, and the result follows.
