= Solution
The sine vectors $v_j^{(m)}=\sin(j\theta_m)$, $\theta_m=m\pi/J$, simultaneously diagonalize these two symmetric <tridiagonal matrices>. Their <eigenvalues> are
$$
M:\quad\frac h3(2+\cos\theta_m),\qquad
K:\quad\frac2h(1-\cos\theta_m).
$$
Thus the semidiscrete decay rates are $\lambda_m=6(1-\cos\theta_m)/[h^2(2+\cos\theta_m)]$. The <forward Euler method> multiplies each mode by $1-k\lambda_m$. <Stability> requires $0\leq k\lambda_m\leq2$ for every mode.
For a fixed finite Dirichlet grid the largest mode is $\theta_{J-1}=\pi-\pi/J$. Consequently the exact finite-grid bound is
$$
\boxed{0\leq\mu\leq\frac{2-\cos(\pi/J)}{3(1+\cos(\pi/J))},\qquad \mu=k/h^2.}
$$
As the mesh is refined, the upper bound decreases to $1/6$. Hence the mesh-independent answer is \b[$\boxed{0\leq\mu\leq1/6}$]. This is the <forward Euler limit for a consistent hat mass matrix>. The frequently seen $1/2$ diffusion bound belongs to the standard second-difference scheme or a lumped <mass matrix>, and is not the bound for the consistent Galerkin equations derived here.
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