Sunday, 15 November 2015

convergence divergence - Show that the sequence given by xn+1=xn+fracsqrt|xn|n2 is convergent



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My Try:



It is clear that xn is monotonically increasing. If we assume that the sequence converges to a then a=a+|a|n2. Hence a=0. So, I was going to prove that the sup of the sequence is 0. But failed. Can somebody please help me to complete the proof


Answer



xn+1=(xn+12n2)214n4<(xn+12n2)2


so
xn+1<xn+12n2

hence
xn+1x1<12nk=11n2<π212


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