Convergence of sequence b

Question

Let $b_n$ be a sequence such that $b_{n+1}=\frac{b_n^2+1}{b_n} \ , \ b_1>0$

Is this sequence converging? explain.

I managed to find that the series is monotically incresing, but couldn't show it is not bounded, thus not converging.

Thanks!

Answer 1

In fact the sequence does not have a limit. Suppose for contradiction that it has a limit $L$. Taking limits as $n\rightarrow \infty$ on both sides of the recurrence equation, $L = \frac{L^2+1}{L} = L + \frac{1}{L}$, or $\frac{1}{L} = 0$, which is clearly impossible.