Linked Questions
49 questions linked to/from Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
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.
- 7,284
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(...
user100593
3
votes
3
answers
479
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 $\lim_{n \rightarrow \infty} \sqrt{n} x_n$...
- 507
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 $\...
- 3,173
0
votes
2
answers
305
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'...
- 679
-2
votes
1
answer
454
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 $0$, ...
- 5
3
votes
1
answer
200
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 ...
- 49.8k
2
votes
0
answers
218
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 ...
- 271
1
vote
1
answer
135
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) &...
- 304
-1
votes
1
answer
81
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 $\lim_{n\to\infty} \frac{x_n}{\sqrt{3}/n} = 1$
- 19
1
vote
1
answer
97
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 ...
- 82.5k
1
vote
0
answers
91
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 ...
- 2,417
1
vote
1
answer
43
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 ...
- 480
0
votes
0
answers
48
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 $\...
- 21
31
votes
2
answers
3k
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}\...
- 22.8k