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=1−12+an.
So, given that both an and an−1 are positive, we have
an<an−1⟹12+an>12+an−1⟹1−12+an<1−12+an−1⟹an+1<an.
So, we can indeed conclude that the sequence is monotonic.
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