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

 Messages: [ Previous | Next ]
 Roland Franzius Posts: 586 Registered: 12/7/04
Re: Sequence limit
Posted: Oct 26, 2013 3:13 AM

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.

--

Roland Franzius

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