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Prove for the expression based on limit of sequence of polynomial

Source: 2019 Jozsef Wildt International Math Competition

May 19, 2020
limitintegrationSequencespolynomialalgebra

Problem Statement

Consider the sequence of polynomials P0(x)=2P_0(x) = 2, P1(x)=xP_1(x) = x and Pn(x)=xPn−1(x)−Pn−2(x)P_n(x) = xP_{n-1}(x) - P_{n-2}(x) for n≥2n \geq 2. Let xnx_n be the greatest zero of PnP_n in the the interval ∣x∣≤2|x| \leq 2. Show that lim⁡n→∞n2(4−2π+n2∫xn2Pn(x)dx)=2π−4−π312\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}
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