Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 14A / a / Solution

Let $Q_k=P^{(k)}$. Differentiating Legendre's differential equation $k$ times and using the Leibniz rule givesThe induction step follows by differentiating this equation once: the derivative of the coefficient $-2(k+1)x$ contributes the additional term $-2(k+1)Q_k^{\prime }$. ThusFor fixed $\lambda$, this is negative for every sufficiently large $k$.