How to solve $a_{n+1}=a_n+\frac{1}{a_n}$

Asked Sep 16 '16 at 08:40

Active Sep 16 '16 at 08:50

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I know that the limit doesn't exist, using Fixed Point Iteration Method and some appropriate inequalities. I can't think of any way to find the explicit form for $a_n$ from the following recurrence relation. $$a_{n+1}=a_n+\frac{1}{a_n}$$

(If this can be solved, then can $a_{n+1}=ka_n+\frac{l}{a_n}$ be solved?)

edited Sep 16 '16 at 08:50

asked Sep 16 '16 at 08:40

Junghyun Lee

Oops. I meant to say that. (Just edited it.) – Junghyun Lee Sep 16 '16 at 08:50
It is a pitty you did not ask for $a_{n + 1} = \frac{1}{2}(a_{n} + \frac{1}{a_{n}})$. Just for curiosity, have a look at http://math.stackexchange.com/questions/1928503/characteristic-equation-of-a-recurrence-relation/1928507#1928507 – Claude Leibovici Sep 16 '16 at 08:58
Precise asymptotics can be computed, but not a closed form. (And this was asked previously on the site.) – Did Sep 16 '16 at 09:28
Do we know for certain that the closed form doesn't exist? – Junghyun Lee Sep 16 '16 at 09:50

How to solve $a_{n+1}=a_n+\frac{1}{a_n}$

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