Limits on recursive sequence

Question

I have no Idea how to evaluate such limits. I checked this http://mathonline.wikidot.com/evaluating-limits-of-recursive-sequences And few questions here. But I'm still not able to understand.

$x_1=\sqrt3$

$x_{n+1}=\dfrac{x_n}{1+\sqrt{1+x_n^2}}$

As mentioned in that webpage, we need see whether the sequence is increasing or decreasing (I'm not able to prove that either). It's clearly decreasing.

I can't prove if the sequence is converging (but I know it is). And I'm not able to evaluate it by the standard method or "trick"

$\lim\limits_{n\to \infty} 2^nx_n$

I don't know what to do with that $2^n$ or how to shift "L"

Well if the sequence has a limit, it should be a root of the equation $$\frac{x}{1+\sqrt{1+x^2}}=x$$ — K.defaoite, Jun 14 '20 at 14:28
It does.. all (options) have $\pi$ in them. Anyway, how did you get there? — UmbQbify, Jun 14 '20 at 14:30
Oh sorry. I thought you wanted the limit of $x_n$, not $2^n x_n$. — K.defaoite, Jun 14 '20 at 16:43
@Fabio Lucchini, thanks, I couldn't find it... But I don't want that brute force solution. — UmbQbify, Jun 15 '20 at 05:57

score 1 · Accepted Answer · answered Jun 14 '20 at 15:09

1

Well, the benefit of such exercises is limited: you see it, or you don't. Of course, you've got a better chance if you had a good drill in various identities (in this case, trigonometry): if you set $$x_n=\tan\theta_n,$$ you get (after some elementary calculation) $$x_{n+1}=\tan\theta_{n+1}=\tan\frac{\theta_n}2,$$ i.e.$$\theta_{n+1}=\frac{\theta_n}2.$$ Since $\theta_1=\frac{\pi}3,$ you see that $$x_n=\tan\frac\pi{3\cdot2^{n-1}},$$ and thus $$\lim_{n\to\infty}2^nx_n=\frac{2\pi}3.$$

answered Jun 14 '20 at 15:09

I was hoping that I'll get a simpler equation, then I'll substitute. Thanks! – UmbQbify Jun 14 '20 at 15:13
But, how can you be sure that the sequence exists in terms of $\theta$ or did you just assume that it exists? – UmbQbify Jun 14 '20 at 15:15
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You can't be sure (in advance), you can but try. – Jun 14 '20 at 15:19

Limits on recursive sequence

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