quasi
Posts:
11,740
Registered:
7/15/05


Re: Sequence limit
Posted:
Oct 3, 2013 5:42 PM


konyberg wrote: >quasi wrote: >> quasi wrote: >> >konyberg wrote: >> >>Bart Goddard wrote: >> >>> >> >>> 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/(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? > >Yes I was a bit hasty here. But sin(n) is limited from 1 to +1 >(your function isn't limited),
Sure it is.
For positive integers n, the function
f(n) = 1/(2^n)
satisfies 0 < f(n) <= 1/2
>and the limit of 1/n is 0. I would think that the limit is 1.
I agree that in this case, the limit is probably equal to 1,
But in general, if f,g are functions such that, for all positive integers n,
(1) 0 < f(n) <= 1 (2) 0 < g(n) (3) g(n) > 0 as n > oo
the question as to whether or not the limit, as n > oo, of
f(n)^g(n)
exists, and if so, to what value, cannot be answered without more information about the functions f,g.
In particular, for this question, it doesn't matter in the least whether the expression 0^0 is regarded as either
undefined equal to 1 equal to 0 equal to 1/2 (hey, split the difference) equal to some other constant
>The debate will still be what 0^0 is equal to :)
Which has no relevance to the OP's question.
quasi

