= Solution
Write $A=D+L+U$, with $D$ diagonal and $L,U$ the strict triangular parts. The <Jacobi method> updates all coordinates using the preceding iterate:
$$
\boxed{x^{(k+1)}=D^{-1}(b-(L+U)x^{(k)}).}
$$
For a symmetric positive definite <matrix>, $D$ has positive diagonal entries. In its splitting take $M=D$, $N=D-A$. The theorem requires $M^*+N=2D-A$ to be positive definite, which need not hold for every positive definite <matrix>.
For this tridiagonal <matrix>, let $S=\operatorname{diag}((-1)^1,\ldots,(-1)^n)$. Neighboring indices have opposite signs, so $SAS=2D-A$: the diagonal stays fixed and every nonzero off-diagonal entry changes sign. Thus
$$
x^T(2D-A)x=(Sx)^TA(Sx)>0\quad(x\ne0).
$$
The <Householder-John theorem> now applies and proves \b[Jacobi convergence for every initial vector]. This is precisely why the tridiagonal structure matters.
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