Home > Olympiad > IMC 2011 Day 2 Problem 1

IMC 2011 Day 2 Problem 1


Let (a_n)\subset (\frac{1}{2},1). Define the sequence x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}. Is this sequence convergent? If yes find the limit.

Solution: Prove by induction that x_n \in (0,1) then prove that (x_n) is increasing (substract two neighboring terms and use the recurrence relation). Therefore (x_n) is convergent. Since (a_n) is bounded it has a convergent subsequence. Use this subsequence and the recurrence relation to prove that the only possible limit point for (x_n) is 1.

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