calculus - Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+frac{1}{a_{n-1}}$ converge?

Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?

Since the function $x+1/x$ is strictly monotonic increasing for all $x>1$, I don't think that the limit converges, but I'm not sure. Can anybody tell me whether the sequence is converging or not?

Answer

No. If it were convergent to some $\alpha$, this value would verify

$$\alpha=\alpha+\frac{1}{\alpha}.$$