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

 Messages: [ Previous | Next ]
 karl Posts: 397 Registered: 8/11/06
Re: Sequence limit
Posted: Oct 26, 2013 3:51 AM

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!

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