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

 Messages: [ Previous | Next ]
 Karl-Olav Nyberg Posts: 1,575 Registered: 12/6/04
Re: Sequence limit
Posted: Oct 3, 2013 4:49 PM

On Thursday, October 3, 2013 10:26:13 PM UTC+2, quasi wrote:
> quasi wrote:
>

> >konyberg wrote:
>
> >>Bart Goddard wrote:
>
> >>>
>
> >>> This question from a colleague:
>
> >>>
>
> >>> What is lim_{n -> oo} |sin n|^(1/n)
>
> >>>
>
> >>> where n runs through the positive integers.
>
> >>>
>
> >>> Calculus techniques imply the answer is 1.
>
> >>> But the same techniques imply the answer is 1
>
> >>> if n is changed to x, a real variable, and that
>
> >>> is not the case, since sin x =0 infinitely often.
>
> >>>
>
> >>> Anyone wrestled with the subtlies of this problem?
>
> >>>
>
> >>> E.g., can you construct a subsequence n_k such
>
> >>> that sin (n_k) goes to zero so fast that the
>
> >>> exponent can't pull it up to 1?
>
> >>
>
> >>In general a^0 = 1. lim (n goes inf) 1/n = 0. Then the value
>
> >>of sin(n) doesn't change that a^0 = 1.
>
> >
>
>
> >
>
> >Let f(n) = 1/(2^n).
>
> >
>
> >Then f(n)^(1/n) = 1/2 for all nonzero values of n, hence the
>
> >limit, as n approaches infinity, of f(n)^(1/n) is 1/2, not 1.
>
> >
>
> >Are there infinitely many positive integers n such that
>
> >
>
> > |sin(n)|^(1/n) < 1/(2^n)
>
>
>
> I meant:
>
>
>
> Are there infinitely many positive integers n such that
>
>
>
> |sin(n)| < 1/(2^n)
>
>
>

> >??
>
> >
>
> >If so, then the limit of the sequence
>
> >
>
> > |sin(n)|^(1/n), n = 1,2,3, ...
>
> >
>
> >does not exist. In particular, it would not be equal to 1.
>
> >
>
> >In fact, the original question can be recast as:
>
> >
>
> >Does there exist a real number c with 0 < c < 1 such that
>
> >the inequality
>
> >
>
> > |sin(n)| < c^n
>
> >
>
> >holds for infinitely many positive integers n?
>
>
>
> quasi

Hi.

Yes I was a bit hasty here. But sin(n) is limited from -1 to +1 (your function isn't limited), and the limit of 1/n is 0. I would think that the limit is 1. The debate will still be what 0^0 is equal to :)

KON

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