recursive succession $a_{n+1}= - \frac{1}{2} (a_n + \frac{3}{a_n})$

Question

I'm given this recursive succession:

$a_{n+1}= - \frac{1}{2} (a_n + \frac{3}{a_n})$

I have to find the limit.

if $a_0 >0$ then $a_1 <0$, $a_2 >0$ ...

if $a_0 <0$ then $a_1 >0$, $a_2 <0$ ...

The sign is alternative and we can't say that the succession is monotone.

In this case I can't say that the limit exists but I don't know how to procede.

Answer 1

If the limit exists, call it $x$, it must satisfy $x=\frac{-1}{2}(x+3/x)$. So $x^2=-\frac{1}{2}(x^2+3)$. Does this have a real solution?

Answer 2

If the limit is equal to $a$ so $a^2=-1,$ which is impossible.

Answer 3

In many of similar problems, you suppose that $\lim_{n\to\infty}a_n=a$ for some $a$, and try to derive a contradiction. In this case, rewrite the recurrence relation as follows:

$$2a_{n+1}a_n+a_n^2=-3$$

which, by letting $a_n\to a$ as $n\to\infty$ means that:

$$3a^2=-3$$

which has no solutions.