= Convergence of an adjacent-term averaging recurrence
{title2=$x_{n+2}=\alpha x_{n+1}+(1-\alpha)x_n$}
For $0<\alpha<1$, the successive differences satisfy $d_{n+1}=-(1-\alpha)d_n$. Summing the resulting <geometric series> gives
$$
x_n=x_0+\frac{x_1-x_0}{2-\alpha}\bigl[1-(\alpha-1)^n\bigr],\qquad
\lim_{n\to\infty}x_n=\frac{(1-\alpha)x_0+x_1}{2-\alpha}.
$$
Equivalently, 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.
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