Wednesday, 4 December 2013

real analysis - Upper bound of a recursive sequence for a fixed n



I have a recursive sequence given by xn=1+xn12, x1=0


I can easily show that it is increasing, bounded and thus converges with its limit being 1. But what if I wanted to upper bound xn0 for some fixed n0? This bound should be dependent on n. How should I proceed?


Answer



x0=cosπ2x1=cosπ4xn=cos2n1π


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