This is a good question, and I happen to have thought about it. Several comments pointed out that for tridiagonal Toeplitz matrix there are other better algorithms; that's true, but it does not answer the question.

I think the confusion comes from the statement "Gauss-Seidel convergence is given by its spectral radius", which is incorrect. Gauss-Seidal convergence depends on the spectral radius of a new matrix L^{-1}R where L-R = A, for solving Ax=b. It is hard to estimate the spectral radius of L^{-1}R, but I am pretty sure that it is related to the condition number of A (the ratio between the largest and the smallest eigenvalues).

This is a good question, and I happen to have thought about it. Several comments pointed out that for tridiagonal Toeplitz matrix there are other better algorithms; that's true, but it does not answer the question.

I think the confusion comes from the statement "Gauss-Seidel convergence is given by its spectral radius", which is incorrect. Gauss-Seidel convergence depends on the spectral radius of a new matrix $L^{-1}R$ where $L-R = A$, for solving $Ax=b$. It is hard to estimate the spectral radius of $L^{-1}R$, but I am pretty sure that it is related to the condition number of $A$ (the ratio between the largest and the smallest eigenvalues).