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Show that the sequence has finite limits - [VMO 2011]

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January 11, 2011
limitalgebra proposedalgebra

Problem Statement

Let ⟨xn⟩\langle x_n\rangle be a sequence of real numbers defined as x1=1;xn=2n(n−1)2∑i=1n−1xix_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i Show that the sequence yn=xn+1−xny_n=x_{n+1}-x_n has finite limits as n→∞.n\to \infty.
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