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

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 Timothy Murphy Posts: 657 Registered: 12/18/07
Re: Sequence limit
Posted: Nov 6, 2013 8:30 AM
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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 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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