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

 Messages: [ Previous | Next ]
 Robin Chapman Posts: 412 Registered: 5/29/08
Re: Sequence limit
Posted: Oct 4, 2013 11:47 AM

On 03/10/2013 19:01, Bart Goddard wrote:
>
> This question from a colleague:
>
> What is lim_{n -> oo} |sin n|^(1/n)
>
> where n runs through the positive integers.
>
> Calculus techniques imply the answer is 1.

I can't believe you said that, Bart!

> Anyone wrestled with the subtlies of this problem?
> E.g., can you construct a subsequence n_k such
> that sin (n_k) goes to zero so fast that the
> exponent can't pull it up to 1?

The limit is 1.

Note that |sin n| = |sin(n - m pi) |
where m is the nearest integer to n/pi.
Then
|sin n| = a_n |n - m pi|
where a_n is between 1 and 2/pi so a_n^{1/n} -> 1.
Also m^(1/n) -> 1 so that the sequence |sin n|^{1/n}
has the same behaviour as |n/m - pi|^{1/n}.

But pi has a finite irrationality measure: there are A > 0
and r with |n/m - pi| > A/m^r for all positive integers m and n.
This means that |n/m - pi|^{1/n} -> 1.

For a nice article on the irrationality measure of pi and other
numbers, see Zudilin's essay:
http://uk.arxiv.org/abs/math/0404523 .

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 Bart Goddard
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