Convergence of an adjacent-term averaging recurrence ($x_{n+2}=\alpha x_{n+1}+(1-\alpha )x_n$)

For $0<\alpha <1$, the successive differences satisfy $d_{n+1}=-(1-\alpha )d_n$. Summing the resulting geometric series givesEquivalently, the two-dimensional update matrix has eigenvalues $1$ and $\alpha -1$, with only the second mode decaying. At $\alpha =0$, generic initial values instead alternate and do not converge.