Does $\displaystyle \{ |\sin n|^{\frac{1}{n}}\}$ converge?

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- Aug 27th 2010, 05:57 PMelimDetermine whether a sequence converge
Does $\displaystyle \{ |\sin n|^{\frac{1}{n}}\}$ converge?

- Aug 27th 2010, 07:13 PMProve It
Remember that

$\displaystyle -1 \leq \sin{n} \leq 1$ for all $\displaystyle n$, that means

$\displaystyle |\sin{n}| \leq 1$ for all $\displaystyle n$.

I assume since this is a sequence that $\displaystyle n$ only takes on postitive integer values, so that would mean $\displaystyle \frac{1}{n}$ is positive and therefore exponentiating would keep the inequality intact.

So $\displaystyle |\sin{n}|^{\frac{1}{n}} < 1^{\frac{1}{n}}$

$\displaystyle |\sin{n}|^{\frac{1}{n}} < 1$.

So yes, the sequence converges. - Aug 27th 2010, 07:23 PMelim
$\displaystyle |\sin n|^{\frac{1}{m}}<1$ not necessarily implies that it converges

- Aug 27th 2010, 07:28 PMProve It
True, I misinterpreted the question, sorry. All I proved was that it was bounded.

According to Wolfram,

$\displaystyle \lim_{n \to \infty}(|\sin{n}|^{\frac{1}{n}}) = 1$

so yes, the sequence converges. - Aug 27th 2010, 08:29 PMelim
Thanks a lot Sir!

what is the url at Wolfram? - Aug 28th 2010, 04:39 AMchisigma
Is...

$\displaystyle \displaystyle |\sin n|^{\frac{1}{n}} = e^{\frac{\ln |\sin n|}{n}$ (1)

... and because ...

$\displaystyle \displaystyle \lim_{n \rightarrow \infty} \frac{\ln |\sin n|}{n}= 0 $ (2)

... is ...

$\displaystyle \displaystyle \lim_{n \rightarrow \infty} |\sin n|^{\frac{1}{n}} = 1$ (3)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - Aug 29th 2010, 09:37 PMelim
$\displaystyle \displaystyle \lim_{n \rightarrow \infty} \frac{\ln |\sin n|}{n}= 0$

is not obvious. there are integer $\displaystyle n's$ that make $\displaystyle |\sin n|$ very close to $\displaystyle 0$ hence $\displaystyle -\ln |\sin n|$ is very big. - Aug 30th 2010, 01:33 AMqspeechc
Do we have $\displaystyle |\text{sin} n|\leq n \quad \forall n\in\mathbb{N}$ ?

- Aug 30th 2010, 02:55 AMAckbeet
Reply to Elim at Post # 5:

Prove It is most likely referring to WolframAlpha. It's basically a way to enter single Mathematica commands. At least, that's what I use it for. Quite handy. - Aug 30th 2010, 05:33 AMelim
- Aug 30th 2010, 05:35 AMelim
- Aug 30th 2010, 10:05 AMchisigma
The consideration of elim is quite correct. By definition say that...

$\displaystyle \displaystyle \lim_{n \rightarrow \infty} \frac{\ln |\sin n|}{n}= 0$ (1)

... means that for any $\displaystyle \varepsilon >0$ it exists an integer $\displaystyle N$ so that $\displaystyle \forall n>N$ is...

$\displaystyle \displaystyle \frac{|\ln |\sin n||}{n}< \varepsilon $ (2)

Now if we measure angles in degree [for example] instead of in radians $\displaystyle \forall n|180$ is $\displaystyle \sin n =0$ and the sequence $\displaystyle \displaystyle a_{n} = |\sin n|^{\frac{1}{n}}$ has no limit. If the angles are in radians then $\displaystyle \sin n$ never vanishes and in my opinion the (1) is 'probably true', even if for $\displaystyle \varepsilon$ 'very small' the (2) is satisfied for 'enormous' values of $\displaystyle N$. In any case a rigorous prove has to be supplied and that is not a trivial job (Thinking)...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - Aug 30th 2010, 11:17 AMqspeechc
- Aug 30th 2010, 12:23 PMDefunkt
- Aug 31st 2010, 08:27 AMqspeechc