Asymptotics for Some Logistic Maps and the Renormalization Group

The paper proves, for the r=1 logistic recurrence λ_{n+1}−λ_n=−βλ_n^2 with 0<λ_0<1/β and β>0, that lim_{n→∞} nλ_n = 1/β. It does so via a discrete “FTC” identity 1/λ_{n+1}−1/λ_n = β/(1−βλ_n), telescoping sums, and bounds on auxiliary sums S_1 and S_{≥2}, culminating in Theorem 2 stating lim nλ_n=1/β (see the setup and telescoping in Eq. (7), (13)–(14), and the conclusion in Theorem 2) . The candidate model solution is also correct: it proves λ_n↓0, sets a_n=1/λ_n, observes Δa_n=β/(1−βλ_n)→β, and applies the elementary (Cesàro/Stolz-type) lemma “if Δa_n→L, then a_n/n→L,” yielding nλ_n→1/β. This is a valid, shorter argument that does not rely on the paper’s logarithmic correction estimates; it obtains the same limit under the same hypotheses. Therefore both are correct, but they follow different proof paths (paper: discrete FTC plus bounds; model: convergent first-difference lemma).