= Solution
A <stationary iterative method for a linear system> converges from every initial vector exactly when the <spectral radius> of its <iteration matrix> is less than one. Here $\cos(kx)$ ranges between $\cos x$ and $-\cos x$, so all modes decay for a fixed $n$ exactly when
$$
-1<1-\omega-\omega\cos x
\leq\lambda_k(\omega)
\leq1-\omega+\omega\cos x<1.
$$
Requiring this for every grid size gives
$$
\boxed{0<\omega\leq1.}
$$
Indeed, $\omega\leq0$ leaves or amplifies the lowest mode, while any $\omega>1$ makes the highest-mode eigenvalue tend to $1-2\omega<-1$ as $n\to\infty$.
The slowest mode is $k=1$, for which the <Taylor expansion> is
$$
\lambda_1(\omega)
=1-\omega+\omega\cos\frac{\pi}{n+1}
=1-\frac{\omega\pi^2}{2(n+1)^2}
+O(n^{-4}).
$$
If the initial error has a nonzero $\mathbf v_1$ component, that component is multiplied by $\lambda_1(\omega)^\nu$. Hence for a constant $c>0$ independent of $n$,
$$
\boxed{\|\mathbf e^{(\nu)}\|
\ \text{cannot in general decay faster than}\
\left(1-\frac{c}{n^2}\right)^\nu.}
$$
This is the low-frequency obstruction to grid-independent convergence.
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