Thread: showing a sequence converges

Can I just check, am I right to say that there are two fixed points, $\displaystyle \pm \sqrt{10}$ and the positive point is stable and the negative is unstable. Because I have found that if you chose any initial value other than $\displaystyle -\sqrt{10}$ then the sequence convergres to $\displaystyle \sqrt{10}$?

I'm assuming you meant $\displaystyle \pm\sqrt{2}$, for the original problem.

It all depends on how rigorous a proof you want. The word "show" has different meanings in different contexts: a pure mathematician means something different by it, sometimes, than an applied mathematician, who means something different from a physicist. If you want to prove, beyond the shadow of a doubt, that the limit is as you've described, you need to do something at least as rigorous as what I've outlined, if not more. If throwing in a few numbers and turning the crank is enough, then just do that. I wouldn't call that a proof by any stretch of the imagination. For one thing, it doesn't take into account the infinite sequence of numbers that are not in the domain of the iteration scheme. Secondly, how do you know it converges to exactly $\displaystyle \sqrt{2}$? A computer has a finite representation of a number with essentially infinite precision. How do you know you're not off in the millionth decimal place?

The lengthy computation I went through does "show" what you're trying to prove. And you might be able to get the subsequence approach to work as well. Either of those is definitely more rigorous than simply plugging in numbers. However, I'm currently working as an engineer, and I know that sometimes the best, most elegant mathematical answer to a problem is not feasible because it takes too long to get. So, if it doesn't work for you, do something else! From my perspective, the long computation or the alternating subsequence approach are the only thing that come to my mind at the moment.

Good luck!

sorry yes, in the post above I mean 2 not 10. Am I right in terms of stable and unstable? And if i were to need just a brief reason why this happens, would it make sense to see the upper and lower bound are $\displaystyle \pm \sqrt{2}$ and so when the sequence increases it will converge to the positive root and when it is decreasing it will be converging to the negative root?

Thanks so much for all your help!

You are correct in terms of the stable fixed points, and the unstable fixed points. However, your analysis beyond that seems incorrect to me. Try this: plot $\displaystyle a_{n}$ versus $\displaystyle n$ for about 10 iterations. Pick $\displaystyle a_{0}=1$, and then do it all again for $\displaystyle a_{0}=2$. You can do this really fast in Excel, although I'm not sure its plotting resolution is the best. So one of those starts lower than the stable fixed point, and the other higher. You'll see that your statements like "when the sequence increases it will converge to the positive root and when it is decreasing it will be converging to the negative root" are incorrect. In reality, I say that all values of $\displaystyle a_{0}$ not equal to $\displaystyle -\sqrt{2}$ or any of the values not in the domain of the iteration scheme will converge (neither all increasing nor all decreasing) to $\displaystyle \sqrt{2}$. It'll be like a spring oscillating about its equilibrium point, until it finally stops oscillating.

ah ok i see what you mean, is there any way of giving a brief reason for convergence in a few lines? I am trying to conclude a report about it and he said it only needs a small explanation?

Small doesn't necessarily mean there isn't a lot of work behind a line of reasoning. My Ph.D. thesis wasn't very long, but it took much longer to write one line of that, than it takes the average humanities person to write a line of his thesis. It's denser. If you've done the work, you can write those lovely words every reader hates to read, "It can be shown that..." I don't know how much work it would be to look at every other member of the sequence, but that might be a lot less work, if you can show that subsequence is monotone. Then you just invoke the theorem your lecturer gave you for each of the subsequences. Done. That shows that the sequence converges, for most values. To show it converges to the limit we know it converges to, is another matter. What exactly do you need to show? Convergence for most of the real line? Or do you have to exhibit an actual value?

It just says to use numerical experimentation to investigate how the iterations of $\displaystyle y=\frac{x+10}{x+1}$ converge. Then try to prove your findings briefly (look up some standard tests for convergence.

It sounds like you've done the numerical experimentation. So that's fine.

I just had another idea: revisit the ratio test. The ratio test says to compare a value with its previous value in ratio. Let

$\displaystyle L=\left|\frac{a_{n+2}}{a_{n+1}}\right|=\left|\frac {3a_{n}+4}{2a_{n}+3}\cdot\frac{a_{n}+1}{a_{n}+2}\r ight|=\left|\frac{3a_{n}^{2}+7a_{n}+4}{2a_{n}^{2}+ 7a_{n}+6}\right|$.

Perhaps there's a direction you could go with this - I don't know. I'm just throwing it out there. I have to get going for the day. Perhaps I'll check in again later.

I would recommend you look into the approach where you take the alternating subsequences. I think you'd find that those subsequences are monotone. It might not even be all that difficult to prove that. If so, you can prove convergence pretty quickly.