TITLE: How fast does the sequence $b_{n+1}=b_n+\frac{1}{b_n}$ grow?
QUESTION [5 upvotes]: $b_{n+1}=b_n+\frac{1}{b_n}$, where $b_0=1$. 
I have proven that this sequence is $O(\ln n)$ and $o(n)$.
Can anyone give a more precise estimation of the order of $b_n$?
i.e. a $f(n)$ such that $0<\lim_{n \to \infty}\frac{b_n}{f(n)}<\infty$

REPLY [2 votes]: Another idea (that needs some additional work).
Approximate your discrete sequence $b_n$ using a continuous function $\beta(x)$:
$$
b_{n+1}-b_n = \frac{1}{b_n}, \qquad b_0=1\tag{1}
$$
$$
\beta'(x) = \frac{1}{\beta(x)},\qquad \beta(0)=1\tag{2}
$$
with solution
$$
\beta(x) = \sqrt{2x+1}
$$
Then what remains is to determine how well the solution of (2) approximates the solution of (1).  Perhaps using the mean value theorem.