= Solution
Split $A=D+L+U$ with $D=\mu^{-1}I$, $L$ the negative unit subdiagonal and $U$ the positive unit superdiagonal. The <Jacobi method> uses $x^{(r+1)}=-D^{-1}(L+U)x^{(r)}+D^{-1}b$, so
$$
\boxed{T_J=\begin{pmatrix}0&-\mu&0&0\\\mu&0&-\mu&0\\0&\mu&0&-\mu\\0&0&\mu&0\end{pmatrix}}.
$$
The <Gauss-Seidel method> uses $(D+L)x^{(r+1)}=b-Ux^{(r)}$, giving by forward substitution
$$
\boxed{T_{GS}=\begin{pmatrix}0&-\mu&0&0\\0&-\mu^2&-\mu&0\\0&-\mu^3&-\mu^2&-\mu\\0&-\mu^4&-\mu^3&-\mu^2\end{pmatrix}}.
$$
Both iterations converge for every initial error exactly when their <spectral radius> is below one, since the error evolves as powers of the iteration matrix.
For Jacobi, the leading tridiagonal determinant recurrence is $p_j(\lambda)=\lambda p_{j-1}(\lambda)+\mu^2p_{j-2}(\lambda)$, with $p_0=1,p_1=\lambda$. Thus
$$
p_4=\lambda^4+3\mu^2\lambda^2+\mu^4.
$$
Its roots are $\lambda=\pm i\mu\sqrt{(3+\sqrt5)/2}$ and $\pm i\mu\sqrt{(3-\sqrt5)/2}$. Expanding the Gauss-Seidel determinant gives
$$
\det(\lambda I-T_{GS})=\lambda^2(\lambda^2+3\mu^2\lambda+\mu^4),
$$
so its nonzero eigenvalues are $-\mu^2(3\pm\sqrt5)/2$. Therefore
$$
\rho(T_J)=\frac{1+\sqrt5}{2}|\mu|,\qquad\rho(T_{GS})=\frac{3+\sqrt5}{2}\mu^2=\rho(T_J)^2.
$$
The <skew-tridiagonal Jacobi and Gauss-Seidel iterations> consequently have the same convergence range:
$$
\boxed{0<|\mu|<\frac{\sqrt5-1}{2}}.
$$
Equality fails the every-start convergence condition, and larger magnitudes have unstable modes. The nonsymmetric matrix does not permit an unexamined positive-definite-matrix convergence argument.
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