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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: Sequence limit
Posted: Oct 3, 2013 4:33 PM

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)

??

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

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