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Topic: Sequence limit
Replies: 72   Last Post: Nov 26, 2013 12:07 AM

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 Leon Aigret Posts: 31 Registered: 12/2/12
Re: Sequence limit
Posted: Oct 4, 2013 11:37 AM
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On Thu, 03 Oct 2013 16:48:16 -0500, quasi <quasi@null.set> wrote:

>quasi wrote:
>>quasi wrote:

>>>>Bart Goddard wrote:
>>>>>
>>>>> This question from a colleague:
>>>>>
>>>>> What is lim_{n -> oo} |sin n|^(1/n)
>>>>>
>>>>> where n runs through the positive integers.

>>>In fact, the original question can be recast as:
>>>
>>>Does there exist a real number c with 0 < c < 1 such that
>>>the inequality
>>>
>>> |sin(n)| < c^n
>>>
>>>holds for infinitely many positive integers n?

>
>I suspect the answer is no.

Assume that such a c exists and that the inequality holds for
infinitely many n..

For such an n there is an integer k for which |k pi - n| <= pi/2 and

therefore |k pi - n| <= pi/2 |sin n| < 2 c^n.

From |pi - n/k| < 2/k c^n it follows that n/k converges towards pi and
that for sufficientlly large n one has n > 2 k, which leads to

|pi - n/k| < 2/k c^(2k).

Simple analysis shows that for any m > 1 the expression 2/x c^(2x) x^m
converges with limit 0 when x -> oo, so for sufficiently large n (and
k) one has

2/k c^(2k) k^m <1 and |pi - n/k| < 1/k^m

In other words, pi would have an infinite irrationality measure.
However, the MathWorld page about the irrationality measure, among
others, mentions a finite upper bound for the irrationality measure of
pi, so the initial assumption does not hold.

Leon

Date Subject Author
10/3/13 Bart Goddard
10/3/13 Karl-Olav Nyberg
10/3/13 quasi
10/3/13 quasi
10/3/13 Karl-Olav Nyberg
10/3/13 quasi
10/4/13 Roland Franzius
10/4/13 quasi
10/5/13 Roland Franzius
10/5/13 quasi
10/26/13 Roland Franzius
10/26/13 karl
10/26/13 Roland Franzius
10/26/13 gnasher729
10/27/13 karl
10/3/13 quasi
10/4/13 Leon Aigret
10/4/13 William Elliot
10/4/13 quasi
10/4/13 William Elliot
10/4/13 quasi
10/4/13 David C. Ullrich
10/4/13 Robin Chapman
10/5/13 Bart Goddard
10/4/13 GoogleOnly@mpClasses.com
10/4/13 Bart Goddard
10/4/13 GoogleOnly@mpClasses.com
10/4/13 Peter Percival
10/5/13 Virgil
10/4/13 Bart Goddard
10/6/13 David Bernier
10/6/13 Virgil
10/6/13 Bart Goddard
10/7/13 Mohan Pawar
10/7/13 Bart Goddard
10/7/13 gnasher729
10/7/13 Richard Tobin
10/7/13 Robin Chapman
10/7/13 Michael F. Stemper
10/7/13 Michael F. Stemper
10/7/13 David Bernier
10/7/13 fom
10/8/13 Virgil
10/8/13 fom
10/8/13 Virgil
10/8/13 fom
10/4/13 fom
10/4/13 quasi
10/4/13 quasi
10/9/13 Shmuel (Seymour J.) Metz
10/10/13 Bart Goddard
11/5/13 Shmuel (Seymour J.) Metz
11/6/13 Bart Goddard
11/11/13 Shmuel (Seymour J.) Metz
11/12/13 Bart Goddard
11/15/13 Shmuel (Seymour J.) Metz
11/15/13 Bart Goddard
11/6/13 Timothy Murphy
11/8/13 Bart Goddard
11/8/13 Paul
11/8/13 Bart Goddard
11/9/13 Paul
11/9/13 quasi
11/9/13 quasi
11/9/13 quasi
11/13/13 Timothy Murphy
11/13/13 quasi
11/14/13 Timothy Murphy
11/14/13 Virgil
11/14/13 Roland Franzius
11/26/13 Shmuel (Seymour J.) Metz
11/9/13 Roland Franzius
11/9/13 Paul

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