Monday, 26 January 2015

calculus - Prove a1=1,an+1=frac1+an2+an converges



I need to prove that the sequence defined by

a1=1,an+1=1+an2+an
converges.



I tried to prove that it's bounded and monotonically decreasing, but I couldn't prove it's monotonically decreasing.
I also managed to find the limit assuming it converges.


Answer



Note that
an+1=1+an2+an=112+an.
So, given that both an and an1 are positive, we have
an<an112+an>12+an1112+an<112+an1an+1<an.
So, we can indeed conclude that the sequence is monotonic.


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