Prove this recursively defined sequence goes to 0

[Ryker]

Homework Statement

Show that $2^{n^{1001}} |a_{n} - a_{\infty}| \rightarrow 0$ as $n \rightarrow \infty.$

Here, a_n is defined recursively by $a_{1} = 1, a_{n+1} = \frac{1}{2}(a_{n}+\frac{x}{a_{n}}).$

I already know that $a_{\infty} = \sqrt{x}.$

Homework Equations

We are given a hint to consider $(y_{n}) = \frac{a_{n} - a_{\infty}}{a_{n} + a_{\infty}}.$

The Attempt at a Solution

I considered the hint, and by defining the sequence in the hint to be (y_n) I got that y_n+1 = y_n². However, I don't know how to proceed and show that the above sequence goes to zero. I know $a_{n} - a_{\infty} \rightarrow 0$ by definition, but I don't see how that y_n thing helps. Any thoughts?

 

[Ryker]

Anyone?

 

[Ryker]

OK, one final bump, and then I give up. But from the lack of responses I see I'm not the only one really puzzled by this problem.

Basically, I need to find a way to show (a_n) as defined above converges faster than (2ⁿ)¹⁰⁰¹ diverges.

 

[Mr.Miyagi]

You haven't mentioned the domain of the functions in $(a_n)$. It should probably not contain 0, as I think the quantity in question doesn't converge for $x=0$.

I haven't checked it, but I assume that $y_{n+1}=y_n^2$ is correct. This implies that $y_{n}=y_1^{2^n}$. Moreover, we can calculate that $|y_1|< 1$ from the definition of $(y_n)$ and the fact that $x\neq 0$. Now $$\begin{eqnarray} 2^{n^{1001}} |a_{n} - a_{\infty}| &=& 2^{n^{1001}} |a_{n} + a_{\infty}|||y_n| \nonumber \\ &\leq& 2^{n^{1001}} |2\sqrt{x}||y_n| \nonumber \\ &=& 2^{n^{1001}} |2\sqrt{x}||y_1^{2^n}| \nonumber \\ &=& 2^{n^{1001}} |2\sqrt{x}||y_1|^{2^n} \end{eqnarray}.$$
Can you carry on from here to prove that this must converge to 0 pointwise? And if you can, is the convergence uniform?

 

[Ryker]

Hi Mr. Miyagi, and thanks for the response! Even though the homework due date has now already past, I appreciate the input. And in fact, after four hours of wracking my brain on how to do this, I did indeed finally come to the same conclusion as you did, and essentially used the fact that exponentials rise faster than polynomials. I feel really stupid for needing such a long time to figure that out, but eh...

Oh, and the domain is the real numbers, but x must be greater than zero.