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VMO 2015 Problem 1

Source: VMO 2015

July 30, 2016
algebracalculuslimits

Problem Statement

Given a non negative real aa and a sequence (un)(u_n) defined by {u1=3un+1=un2+n24n2+aun2+3 \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases}
a) Prove that for a=0a=0, the sequence is convergent and find its limit.
b) For a∈[0,1]a\in [0,1], prove that the sequence if convergent.
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