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.
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)
??
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

