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Convergent sequence of real numbers

Source: Southern Summer School, gr. 12

July 9, 2017
analysisSequencereal numberConvergence

Problem Statement

Given a real number aa and a sequence (xn)n=1∞(x_n)_{n=1}^\infty defined by: {x1=1x2=0xn+2=xn2+xn+124+a\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right. for all positive integers nn. 1. For a=0a=0, prove that (xn)(x_n) converges. 2. Determine the largest possible value of aa such that (xn)(x_n) converges.
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