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Romania District Olympiad 2009 - Grade XI

Source:

April 10, 2011
limitinductionreal analysisreal analysis unsolved

Problem Statement

Let (xn)n≥1(x_n)_{n\ge 1} a sequence defined by x1=2, xn+1=xn+1n, (∀)n∈N∗x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*. Prove that lim⁡n→∞xn=1\lim_{n\to \infty} x_n=1 and evaluate lim⁡n→∞xnn\lim_{n\to \infty} x_n^n.
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