Linked Questions

4 votes
1 answer
1k views

Need help solving Recursive series defined by $x_1 = \sin x_0$ and $x_{n+1} = \sin x_n$ [duplicate]

$x_1 = \sin x_0 > 0$ $x_{n+1} = \sin x_n$ Prove $\lim_{x \to \infty }$ $\sqrt{\frac{n}{3}} $ $x_n = 1$ having problem of trying to figure out what value for the $x_0$ starts at.
yiyi's user avatar
  • 7,362
1 vote
1 answer
5k views

Prove that $\sin(\sin...(\sin(x))..)$ converges asymptotically to zero [duplicate]

I'm not able to mathematically prove that the equation $$x(k+1)=\sin(x(k))$$ converges asymptotically to zero. By a simple thought it can be concluded that for any $x(0) \in \mathbb{R}$ it applies $x(...
user avatar
3 votes
3 answers
565 views

Proving $\sqrt{n}(x_n)$ converges when $x_n = \sin(x_{n-1}), x_1=1$ [duplicate]

This is a problem that showed up on a qual exam that I have been stuck on for a while. Let \begin{equation} x_n = \sin(x_{n-1}), x_1 = 1 \end{equation} Prove $\lim_{n \rightarrow \infty} \sqrt{n} x_n$...
Story123's user avatar
  • 517
2 votes
1 answer
1k views

Limit of a trigonometric sequence [duplicate]

For an arbitrary $x_{0}$ in $\left(\, 0,\pi\,\right)$ we define $x_{n + 1}=\sin\left(\, x_{n}\,\right)$. Using the limit of the sequence as $n$ tends to infinity we're supposed to find the limit of $\...
Victor's user avatar
  • 3,212
0 votes
2 answers
464 views

Determine the convergence of the recursion $x_n=\sin(x_{n-1})$ [duplicate]

I want to determine the convergence of $$\begin{cases}x_0=1 & \\ x_n=\sin(x_{n-1}) \end{cases}$$ I can see that $x_n=\sin(x_{n-1}) \geq -1$. Which means the sequence is bounded. However, it isn'...
ImHackingXD's user avatar
  • 1,090
-2 votes
1 answer
477 views

$x_1=\sin(x_0)>0, x_{n+1}=\sin(x_n)$, prove $\sqrt{\frac{n}{3}}x_n \to 1$ as $n \to \infty$ [duplicate]

I'm studying convergent sequences at the moment. And I came across this question in the section of Stolz Theorem. I realised that $\{x_n\}$ is monotonously decreasing and has a lower bound of $0$, ...
JJS's user avatar
  • 5
3 votes
1 answer
219 views

How fast does $f_n = \sin f_{n-1}$ approach zero? [duplicate]

The sequence $f(n) = \sin(\sin(\sin(......(1)......)))$ approaches zero like $\sqrt{3/n}$, as has been asked and answered here a few times. So $f(n)$ would get below $1/n$ after $3n^2$ steps, but it ...
Empy2's user avatar
  • 51.2k
2 votes
0 answers
249 views

How do I solve this limit: $ \lim_{x \to \infty } \sqrt{n}\sin(\sin(\sin ... (\sin (1))...)) $ [duplicate]

I have been strugling a lot to solve this question, but couldn't figure out where to start. $$ \lim_{x \to \infty } \sqrt{n} . \underbrace {\sin(\sin(\sin ... (\sin (1))...))}_{n...times..} $$ I ...
user63762453's user avatar
1 vote
1 answer
138 views

iterative sinus [duplicate]

I saw this question online "We define a series $\{a_i\}_{i=0}^\infty$ like so: $a_0 = 1, \; a_{n+1} = sin(a_n)$ prove that $a_n$ converges" that is rather easy because if $\forall x>0 ,\;sin(x) &...
Ofer Magen's user avatar
-1 votes
1 answer
94 views

How I can to prove the succession $x_n$ converge to 1? [duplicate]

Let $0<x_0<1$ if $x_{n+1} = sin(x_n)$ show that $\lim_{n\to\infty} \frac{x_n}{\sqrt{3}/n} = 1$
Sergio MNZ's user avatar
1 vote
1 answer
99 views

Convergence of a sequence with repeated sines [duplicate]

Let $x\in(0,\pi/2)$ and $\{a_n\}_{n\in\mathbb N}$ defined recursively as follows: $$ a_0=x, \quad \text{and} \quad a_{n+1}=\sin(a_n). $$ Show that $$ \lim_{n\to\infty}{n\,a_n^2}=3. $$ Note. There is ...
Yiorgos S. Smyrlis's user avatar
1 vote
0 answers
96 views

Find $\lim_{n\to\infty} \sqrt{n}a_n$ [duplicate]

Define $a_n$ is real sequence which satisfies $$a_1=1, \quad a_{n+1}=\sin(a_n)$$ Find $$\lim_{n\to\infty} \sqrt{n}a_n$$ I just know $$\lim_{n\to\infty} a_n = 0$$ but I don't know what should I ...
bFur4list's user avatar
  • 2,761
1 vote
1 answer
58 views

How slow does sine iteration converges? [duplicate]

It is no hard to prove that the real sequence $\{a_n\}_{n=1}^\infty$ decided by $$ \begin{cases} a_1 = 1\\ a_{n+1} = \sin a_n \end{cases} $$ converges to $0$ as $n\to\infty$. However, it seems that ...
Mr. Egg's user avatar
  • 658
0 votes
0 answers
50 views

Show that there exists a unique real number $\alpha$ such that the sequence $\left(a_{n+1}^{\alpha}-a_{n}^{\alpha}\right)$ converges to some limit. [duplicate]

Let $\left(a_{n}\right)_{n \geqslant 1}$ be the sequence defined by $$ \left\{\begin{array}{l} a_{1}=1 ; \\ \forall n \geqslant 1, a_{n+1}=\sin \left(a_{n}\right) \end{array}\right. $$ (a) Show that $\...
Peter Green's user avatar
30 votes
2 answers
4k views

Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$

I was asked today by a friend to calculate a limit and I am having trouble with the question. Denote $\sin_{1}:=\sin$ and for $n>1$ define $\sin_{n}=\sin(\sin_{n-1})$. Calculate $\lim_{n\to\infty}\...
Belgi's user avatar
  • 23.2k

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