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

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Robin Chapman

Posts: 282
Registered: 5/29/08
Re: Sequence limit
Posted: Oct 7, 2013 1:46 PM
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On 07/10/2013 15:34, Mohan Pawar wrote:

> I won't attempt to attain a low level to match you in ranting and
> displaying vitriol. At times it is better to keep shut and let others
> wonder that you are stupid than to open your mouth and remove their all
> doubts. Bart Goddard, when others critically read the solution under
> "somehow explained to Bart Goddard" you will have removed their doubts
> about you.


Bart might be a bit blunt at times, but he is one of the most
knowledgable contributors remaining in sci.math.

> Let y = |sin (1/m) |^(m) Taking log on both sides => ln(y) = m
> ln(|sin (1/m)|) Take limit on both sides as m->0 and evaluating it
> =>lim m->0 ln(y) = lim m->0 m ln(|sin (1/m)|)= 0 (at the time of
> evaluating limit, m=0 is the multiplier and one doesn?t need to care
> about value of ln(|sin (1/m)|). Also, the limit is determinate.)


You do need to care about the value of log |sin(1/m)|.
Note that here m = 1/n in the original problem, so that
m is a reciprocal of a natural number. With this class of m,
1/m can be arbitrarily close to an integer multiple of pi, and
so |sin(1/m)| can be arbitarily close to zero and log|sin(1/m)|
can be the negative of an arbitrarily large number. The question
is, whether then multiplying log|sin(1/m)| will give you a sequence
tending to zero or not. A priori, |log|sin(1/m)|| might get large
enough so that even m|log|sin(1/m)|| doesn't tend to zero.
(It doesn't but that needs some nontrivial mathematics).

You should realize that if you have two sequences (a_n) converging
to zero and (b_n) then it is not automatic that the sequence
(a_n b_n) converges to zero. To determine it limiting behaviour,
you need more information about both the original sequences.
(Your botched argument is the case a_n = 1/n and b_n = log|sin(1/n)|;
certainly a_n -> 0 but that is not enough by itself to conlcude
a_n b_n -> 0. Note that (b_n) is here an *unbounded* sequence.)

> =>lim m->0 ln(y) = 0 => lim m->0 y = e^0=1 => lim m->0 |sin (1/m)
> |^(m)= 1 as before. (ALSO, VERIFIABLE ON WOLFRAM ALPHA)


Well, if Wolfie Alpha says it's true, it must be true :-)

Returning to the original and less confusing notation then
(i) the sequence (c_n) defined for natural numbers n by
c_n = |sin(n)|^{1/n} converges to 1, and
(ii) the function f(x) defined for real x>0 by
f(x) = |sin(x)|^{1/x} does not have a limit as x -> infinity.

As we have seen, (ii) is easy, but (i) is relatively difficult
to prove.


Date Subject Author
10/3/13
Read Sequence limit
Bart Goddard
10/3/13
Read Re: Sequence limit
Karl-Olav Nyberg
10/3/13
Read Re: Sequence limit
quasi
10/3/13
Read Re: Sequence limit
quasi
10/3/13
Read Re: Sequence limit
Karl-Olav Nyberg
10/3/13
Read Re: Sequence limit
quasi
10/4/13
Read Re: Sequence limit
Roland Franzius
10/4/13
Read Re: Sequence limit
quasi
10/5/13
Read Re: Sequence limit
Roland Franzius
10/5/13
Read Re: Sequence limit
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
Read Re: Sequence limit
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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