If the expression has a limit it must be 1. This is the same as looking for the limit of (1/n)*log |sin(n)|. Since pi is irrational, Weyl's criterion tells us the values of |sin(n)| are uniformly distributed in [0,1]. In particular, there's a positive probability (i.e. the set of n with this property has positive asymptotic density) that its uniformly bounded away from 0, so that the limit over that subsequence must be 0 (and thus the original is 1). The only fly in the ointment are for those n which are close to m*pi for some integer m. That is n/m is a good approximation to pi. We do know about the irrationality measures for pi. There is a positive constant C, and exponent e, so that | pi - n/m | > C/m^e The best known e is due to Hata (see Wadim Zudlin's nice survey http://arxiv.org/abs/math/0404523) and is a little more than 8. Nevertheless, it's thought (conjecture) that the "right" value is e=2 (though this is somewhat controversial). So |m*pi - n| > C/m^(e-1) or we have |sin(n)| is approximately C'/n^(e-1) (for some other C', since n <= 4m). Taking logs says that log |sin(n)| is approximately log C' - (e-1)*log(n). So dividing by n and taking the limit gives 0. So I'd say that the limit in the original problem exists and is 1. Victor On Fri, Dec 27, 2013 at 11:19 AM, Dan Asimov <dasimov@earthlink.net> wrote:
P.S. On the subject, a friend challenged me to determine with proof the limit as n -> oo of |sin(n)|^(1/n), where n takes integer values (or show the limit does not exist).
I haven't solved this yet, but maybe you will.
--Dan
On 2013-12-26, at 10:45 PM, Dan Asimov wrote:
In[1]:= Limit[Abs[Sin[x]]^(1/x), x -> Infinity]
Out[1]= 1
But on the RHS of In[1] inside the limit, it takes the value 0 for arbitrarily large x.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun