Date: Oct 3, 2013 4:56 PM
Author: quasi
Subject: Re: Sequence limit
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.

>>

>>Your logic is flawed.

>>

>>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?

I suspect the answer is no.

I think a comparison of power series might a good way to

attack the problem.

For that approach, I would revise the question as follows:

Does there exist a positive real number c such that the

inequality

|sin(n)| < exp(-c*n)

holds for infinitely many positive integers n?

Alternatively, we could ask this question instead:

Does there exist a positive real number c such that the

inequality

sin^2(n) < exp(-c*n)

holds for infinitely many positive integers n?

quasi