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ln and sequence

Source: 36-th Vietnamese Mathematical Olympiad 1998

February 16, 2007
logarithmsintegrationlimitreal analysisreal analysis unsolved

Problem Statement

Let a≥1a\geq 1 be a real number. Put x1=a,xn+1=1+ln⁡(xn21+ln⁡xn)(n=1,2,...)x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...). Prove that the sequence {xn}\{x_{n}\} converges and find its limit.
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