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

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Roland Franzius

Posts: 436
Registered: 12/7/04
Re: Sequence limit
Posted: Oct 26, 2013 5:17 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Am 26.10.2013 09:51, schrieb karl:
> Am 26.10.2013 09:13, schrieb Roland Franzius:
>> Am 05.10.2013 09:04, schrieb quasi:
>>> Roland Franzius wrote:
>>>> quasi wrote:
>>>>>
>>>>> How do you justify the claim
>>>>>
>>>>> abs(sin(eps_j)) < (1/2)*eps_j
>>>>>
>>>>> ??

>>>>
>>>> We called it the epsilon trick.

>>>
>>> Definitely quite a trick since, for 0 < eps_j < Pi/2, the above
>>> inequality is always false.

>>
>>
>> The conjecture that liminf_{n -> oo} |sin n|^(1/n) = 1 seems more or less plausible from computer experiments.
>>
>> Using Mathematica, I just checked the first 50 continued fraction approximations of pi which give the series of best
>> rational bounds of pi within one unit of the denominator
>>
>> cf(pi,n) = a_n/b_n
>>
>> then
>>
>> log a_n is prop 1.15 n
>>
>> and
>>
>>
>> log|(sin a_n)| is prop - 1.15 n
>>
>> and for n<50
>>
>> 0.5 < a_n * sin a_n < 1.5
>>
>> For n greater than 55 my system with 8 GB gives up.
>>
>> Of course these experiments do not rule out, that very good approximations of pi occur infinitely often, but to me it
>> seems more or less improbable that a subseries a_n_k of approximations giving
>>
>> liminf |sin(a_n_k)|^1/(a_n_k) < 1
>>
>> may exist.
>>
>>
>> See more on
>>
>> http://mathworld.wolfram.com/PiContinuedFraction.html
>>
>> for the series q_n ^(1/n) for the denominators q_n with a seemingly constant limit which according to Eric Weisstein
>> has not yet proven.
>>
>> Here n is the index of the n-th denominator in the truncated continued fraction series. After complete cancellation of
>> common factors, the denominators and numerators grow exponentially with the index.
>>
>>

> This is no answer to quasi's question, clever Dick!
>


We don't give answers to open problems here, poor charly, have fun
yourself.

--

Roland Franzius




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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