Linked Questions

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0answers
24 views

prove that the sequence given by recurrence converges [duplicate]

Given a: $a_0>0$, and $a_{n+1}=\sin(a_n)$. I need to prove that the sequence $a_{n=0}^\infty$ converges, then calculate the limit. I have proved that is crescent, then I got stuck.
1
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1answer
100 views

Repeated applications of cos function converges around $0.7390851$ [duplicate]

I set my calculator to radian mode. I enter any random number $x$. Then I repeatedly apply cosine to that number, getting the sequence $$\cos x, \cos(\cos x), \cos(\cos(\cos x)) \ldots$$ and so on. No ...
0
votes
4answers
61 views

what are the properties of this sequence? [closed]

Can we assert that this sequence is convergent or divergent? $$u_o = a, \text{ and } \forall n \in \mathbb{N} : u_{n+1} = \sin(u_n)$$ What is the limit of $u_{n}$ if a=$\frac{\pi}{4}$
1
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0answers
104 views

Finding value of $\lim_{n\rightarrow \infty}\sin(\sin(\sin(…(\sin \left(\frac{\pi}{2}\right))))$ [duplicate]

Finding value of $\lim_{n\rightarrow \infty}\sin(\sin(\sin(.......(\sin \left(\frac{\pi}{2}\right))))$ Attempt: $\displaystyle \sin(\frac{\pi}{2})=1$ so $\displaystyle \sin(\sin \frac{\pi}{2})= \...
0
votes
1answer
53 views

Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$ [duplicate]

Basically let $S(x)=\sin(x)$ such that $S^2(x)=\sin(\sin(x))$ and $S^3(x)=\sin(\sin(\sin(x)))$ and so on until $S^n(x)=\sin(\sin(\ldots\sin(x)\ldots))$ Prove that $S^n(x)\rightarrow 0$ as $n\...
1
vote
1answer
129 views

A variant of $\lim_{n \to \infty }\underbrace{\sin \sin \dots\sin(t)}_{\text{$n$ compositions}}$

In this question Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$ it's proved that $$\lim_{n \to \infty }\underbrace{\sin \sin \dots\sin(t)}_{\text{$n$ compositions}}, t\in \mathbb {...
1
vote
2answers
76 views

Limit of repeated $\sin$ [duplicate]

I was wondering if it was possible to formalize the following. If we let $f_n(x)$ denote the sequence of function so that $f_1(x)=\sin(x)$, $f_2(x)=\sin(\sin(x))$, $f_3(x)=\sin(\sin(\sin(x)))$ and so ...
0
votes
1answer
454 views

Prove sequence ${x_{n + 1}} = \sin {x_n},{x_1} = 1 $ has a limit

Prove that the sequence defined by $${x_{n + 1}} = \sin {x_n},\ {x_1} = 1$$ has a limit. Ok, I want to prove by Weierstrass: This sequence is monotonically decreasing Sequence is bounded How can ...
2
votes
0answers
33 views

Show the convergent of sequences and limit? [duplicate]

a)Show that the following sequence of functions is convergent. $f_{1}(x) = \sin(x)$ $f_{2}(x) = \sin(\sin(x))$ $f_{3}(x) = \sin(\sin(\sin(x)))$ ... $f_{n}(x) = \sin(\sin(\sin(...(\sin(x))))$ ($n$...
2
votes
2answers
139 views

Convergence of a sequence defined by cos function.

Let $x_{0}=0$. Define $x_{n+1}=cos (x_{n})$ for $n \geq 0$. then prove that sequence $x_{n}$ is convergent and $x_{2n}< \lim_{n\to\infty} x_{n} < x_{2n+1}$ for every $n \in \mathbb{N}$. ...
0
votes
1answer
236 views

Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. Why am I wrong? [duplicate]

Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. It is known that after the first sine, we get something in $[-1,1]$. If it is $0$ then it is constant and ...
0
votes
0answers
75 views

How prove $\frac{1}{\sqrt{2006}}<a_{2006}<\frac{2}{\sqrt{2006}}$ in sequence?

Let $\{a_{n}\}_{n=0}$ be defined recursively by $a_{0}=\pi/4$, and $a_{n+1}=\sin a_{n}$ for $n=0,1,\ldots$. How prove $\frac{1}{\sqrt{2006}}<a_{2006}<\frac{2}{\sqrt{2006}}$?
4
votes
2answers
223 views

Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $$ then I'm done. It's monotonic and decreasing and bounded by 0 ...
0
votes
3answers
299 views

How find this limit $\lim_{n\to\infty}\underbrace{\sin{\sin{\cdots\sin{x}}}}_{n},x\in R$ [duplicate]

Find this limit $$\lim_{n\to\infty}\underbrace{\sin{\sin{\cdots\sin{x}}}}_{n},x\in R$$ My idea: let $$f(x)=\underbrace{\sin{\sin{\cdots\sin{x}}}}_{n}$$ then $$f(x+2\pi)=f(x)$$,so we only consider $x\...
-2
votes
3answers
2k views

Limit of infinite loops of sin x as n tends to infinity [duplicate]

Show that $$lim_{n\to\infty} \text {sin sin ... sin x} = 0 $$ for all x. Note that the n here refers to the number of sin in the expression above.

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