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limit of an interesting sequence

Source: Romanian District Olympiad 2000, Grade XI, Problem 3

September 24, 2018
limitsSequencescalculusreal analysiscontestsromania

Problem Statement

Let be two distinct natural numbers k1 k_1 and k2 k_2 and a sequence (xn)n≥0 \left( x_n \right)_{n\ge 0} which satisfies x_nx_m +k_1k_2\le k_1x_n +k_2x_m, \forall m,n\in\{ 0\}\cup\mathbb{N}.
Calculate lim⁡n→∞n!⋅(−1)1+n⋅xn2nn. \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} .
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