Determine whether a sequence converge

**elim** · August 27th 2010, 05:57 PM

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

**Prove It** · August 27th 2010, 07:13 PM

Remember that

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

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

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

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

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

So yes, the sequence converges.

**elim** · August 27th 2010, 07:23 PM

$|\sin n|^{\frac{1}{m}}<1$ not necessarily implies that it converges

**Prove It** · August 27th 2010, 07:28 PM

True, I misinterpreted the question, sorry. All I proved was that it was bounded.

According to Wolfram,

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

so yes, the sequence converges.

**elim** · August 27th 2010, 08:29 PM

Thanks a lot Sir!
what is the url at Wolfram?

**chisigma** · August 28th 2010, 04:39 AM

Is...

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

... and because ...

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

... is ...

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

Kind regards

$\chi$ $\sigma$

**elim** · August 29th 2010, 09:37 PM

$\displaystyle \lim_{n \rightarrow \infty} \frac{\ln |\sin n|}{n}= 0$
is not obvious. there are integer $n's$ that make $|\sin n|$ very close to $0$ hence $-\ln |\sin n|$ is very big.

**qspeechc** · August 30th 2010, 01:33 AM

Do we have $|\text{sin} n|\leq n \quad \forall n\in\mathbb{N}$ ?

**Ackbeet** · August 30th 2010, 02:55 AM

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.

**elim** · August 30th 2010, 05:33 AM

Originally Posted by qspeechc

Do we have $|\text{sin} n|\leq n \quad \forall n\in\mathbb{N}$ ?

YES

**elim** · August 30th 2010, 05:35 AM

Originally Posted by Ackbeet

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.

Thanks a lot Ackbeet. Very nice to know the online tool.

**chisigma** · August 30th 2010, 10:05 AM

Originally Posted by elim

$\displaystyle \lim_{n \rightarrow \infty} \frac{\ln |\sin n|}{n}= 0$
is not obvious. there are integer $n's$ that make $|\sin n|$ very close to $0$ hence $-\ln |\sin n|$ is very big.

The consideration of elim is quite correct. By definition say that...

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

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

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

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

...

Kind regards

$\chi$ $\sigma$

**qspeechc** · August 30th 2010, 11:17 AM

Originally Posted by elim

YES

I don't mean to be irritating, but if you can show $n^{1/n}\rightarrow 0$ then we are done...

**Defunkt** · August 30th 2010, 12:23 PM

Originally Posted by qspeechc

I don't mean to be irritating, but if you can show $n^{1/n}\rightarrow 0$ then we are done...

$\displaystyle n^{1/n}\rightarrow 1$ :P

**qspeechc** · August 31st 2010, 08:27 AM

Originally Posted by Defunkt

$\displaystyle n^{1/n}\rightarrow 1$ :P

Oops. lol, I must have been smoking something potent...

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