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Reciprocal Sums and Boring Bounds

Source: Indian Statistical Institute Entrance UGB 2023/6

May 14, 2023
inequalitiesrecurrence relation

Problem Statement

Let {un}n≥1\{u_n\}_{n \ge 1} be a sequence of real numbers defined as u1=1u_1 = 1 and un+1=un+1un for all n≥1. u_{n+1} = u_n + \frac{1}{u_n} \text{ for all $n \ge 1$.} Prove that un≤3n2u_n \le \frac{3\sqrt{n}}{2} for all nn.
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