Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?
Note that $a_n^2 = \left(a_{n-1}+\dfrac{1}{a_{n-1}}\right)^2 = a_{n-1}^2 + 2 + \dfrac{1}{a_{n-1}^2} \ge a_{n-1}^2 + 2$.
Therefore, $a_n^2 \ge 2n-1$, and thus, $a_n \ge \sqrt{2n-1}$ for all $n \ge 1$. That's enough to show divergence.
No. If it were convergent to some $\alpha$, this value would verify $$\alpha=\alpha+\frac{1}{\alpha}.$$
Assume it converges, then it does so to a limit $L \ge 1$.
Then we have $L = L + \frac{1}{L}$ which is not possible.