Consider following LTI system $$y[n] - 2y[n-1] = x[n]$$ where $x[n]$ is the input to the system and $y[n]$ is the output. Let $x[n] = \cos[n\frac{\pi}{3}] + 2\cos[n\frac{\pi}{2} + \frac{\pi}{4}]$, determine the Fourier series coefficients for $y[n]$.
My try: First of all we should determine impulse response $h[n]$. So we have $h[n] - 2h[n-1] = \delta[n]$ and assuming initial rest condition leads to $h[n] = 2^nu[n]$. Then the frequency response could be computed as follows: $$H(e^{j\omega}) = \sum_{n = -\infty}^{+\infty}h[n]e^{-jwn} = \sum_{n = 0}^{+\infty}2^ne^{-jwn} = \sum_{n = 0}^{+\infty}(2e^{-jw})^n$$ It's a geometric series and divergent because $|2e^{-j\omega}| = 2 >1$. So we should conclude that response to $x[n]$ doesn't exist? Also is initial rest a valid assumption here?
Edit: Let $a_k$ be the Fourier series coefficients for the input signal $x[n]$. Then the Fourier series coefficients for output is $b_k = a_kH(jk\omega_0)$ where $\omega_0$ is the Fundamental frequency of the input. The problem is that for each $k$ the geometric series diverges so $b_k$ doesn't exist.