karl
Posts:
228
Registered:
8/11/06


Re: Sequence limit
Posted:
Oct 27, 2013 9:31 AM


Am 26.10.2013 10:41, schrieb christian.bau: > On Saturday, October 26, 2013 8:13:31 AM UTC+1, Roland Franzius wrote: > > Excellent and highly relevant post. > > So the situation is this: > > For real numbers anyone but the most inexperienced see immediately that the limit > > lim sin (x)^(1/x) > x>inf > > doesn't exist, because there are arbitrarily large values x where sin (x)^(1/x) is 0, and arbitrarily large values x where sin (x)^(1/x) is 1, > > If x is restricted to integers, then it is obvious that > > lim f (x)^(1/x) > x>inf > > is 1 if f (x) has a lower bound greater than 1 and an upper bound. Only slightly less obvious is that if f (x) has an upper bound and f (x) > 0 for all x (which it is in our case) and we take the subsequence of x's which make f (x) closer to 0 than for any smaller x, and lim f (x)^(1/x) is 1 for that subsequence, then it is 1 for the whole sequence. > > Roland's post demonstrated that it is most likely that the limit is 1. >
Ok, you know the joke, how engineers prove that all odd number are prime? Where is the difference?

