Wednesday, 28 December 2016

real analysis - How does (xn) converge for x1=1, xn+1=frac1xn+3 for n=1,2,ldots?



Show that the sequence (xn) defined by x1=1andxn+1=1xn+3(n=1,2,) converges and determine its limit ?



I try to show (xn) is a Cauchy sequence or (xn) is decreasing (or increasing) and bounded sequence but I fail every step of all.


Answer



Hint: For x,y0 we have |1x+31y+3|=|yx(x+3)(y+3)|19|xy|.


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