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

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Timothy Murphy

Posts: 503
Registered: 12/18/07
Re: Sequence limit
Posted: Nov 6, 2013 8:30 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Bart Goddard wrote:

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


This is an interesting question,
though it is slightly inaccurately stated.
It would be more correct to ask:

Is the sequence |sin n|^1/n convergent?

It's clear that the upper limit is 1,
since n pi mod Z will be distributed evenly in [0,1),
and so will infinitely often be in the range (1/3,2/3).

If the lower limit is < 1, this would imply
that the continued fraction for pi
has infinitely many coefficients a_n of exponential size.
I think one can probably prove that the set of numbers
in a finite range, say [3,4], with this property
has measure 0.
So one could say that a given number is very unlikely
to have the property.

There are simple generalized continued fractions for pi
that do not have exponentially large coefficients,
as given eg in
<http://en.wikipedia.org/wiki/Generalized_continued_fraction>.
But I do not know if there is any theorem stating that
any very close rational approximant to pi
is given by a convergent in such a case.


--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
School of Mathematics, Trinity College, Dublin 2, Ireland



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

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