This is a practice problem from a intro. time series class:
Consider the time series $x_t = \beta_0 + \sum_{i = 1}^k (A_i\cos(2\pi w_i t) + B_i\sin(2\pi w_i t))$, where $\beta_0, w_1, \ldots, w_k$ are constants and $A_1, \ldots, A_k, B_1, \ldots, B_k$ are independent random variables with zero means and variances $\operatorname{Var}(A_i) = \operatorname{Var}(B_i) = \sigma_i^2$. Find the autocovariance function of $x_t$, and prove that $x_t$ is weakly stationary.
Here's what I was thinking: $w_i t$ and $w_i s$ are going to be constants; and, the cosine of $k*2\pi$ is always $0$, which means we can disregard the cosine terms of the sum.
Similarly, each sine term would simply be $B_i$, since the sine of any $k 2\pi$ is always $1$.
But this seems to mean that the time series $x_t$ doesn't actually depend on $t$, but only on what $B_i$'s are.
This means the autocovariance function is:
$$\gamma_x (s, t) = \operatorname{Cov}(x_s, x_t) = \operatorname{Cov}(\beta_0 + B_1 + \cdots + B_k, \beta_0 + B_1 + \cdots + B_k). $$
This doesn't quite seem right. Am I on the right track, or am I doing something wrong? Any guidance would be great. Thank you.