quasi
Posts:
12,050
Registered:
7/15/05


Re: Sequence limit
Posted:
Oct 3, 2013 4:33 PM


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?
quasi

