= Solution
Expanding the order condition about $w=1$ yields $\rho(w)=\sigma_s\sum_{l=1}^s w^{s-l}(w-1)^l/l$ and $\sigma_s=(\sum1/l)^{-1}$. Thus BDF2 has $\rho=w^2-4w/3+1/3$, $\sigma_2=2/3$, and BDF3 has $\rho=w^3-18w^2/11+9w/11-2/11$, $\sigma_3=6/11$. Their first characteristic <polynomials> satisfy the root condition, so consistency plus Dahlquist equivalence gives convergence. For BDF2 the stability boundary $z=\rho(e^{i\theta})/(\sigma_2e^{2i\theta})$ has nonnegative real part; hence the whole left half-plane is stable.
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