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x_{n+1}=f(x_n)

Source: 32-th Vietnamese Mathematical Olympiad 1994

February 24, 2007
trigonometrycalculuscalculus computations

Problem Statement

Define the sequence {xn}\{x_{n}\} by x0=a∈(0,1)x_{0}=a\in (0,1) and xn+1=4π2(cos⁡−1xn+π2)sin⁡−1xn(n=0,1,2,...)x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...). Show that the sequence converges and find its limit.
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