
Re: Sequence limit
Posted:
Nov 6, 2013 8:30 AM


Bart Goddard wrote: > This question from a colleague: > > What is lim_{n > oo} sin n^(1/n) > > where n runs through the positive integers.
This is an interesting question, though it is slightly inaccurately stated. It would be more correct to ask:
Is the sequence sin n^1/n convergent?
It's clear that the upper limit is 1, since n pi mod Z will be distributed evenly in [0,1), and so will infinitely often be in the range (1/3,2/3).
If the lower limit is < 1, this would imply that the continued fraction for pi has infinitely many coefficients a_n of exponential size. I think one can probably prove that the set of numbers in a finite range, say [3,4], with this property has measure 0. So one could say that a given number is very unlikely to have the property.
There are simple generalized continued fractions for pi that do not have exponentially large coefficients, as given eg in <http://en.wikipedia.org/wiki/Generalized_continued_fraction>. But I do not know if there is any theorem stating that any very close rational approximant to pi is given by a convergent in such a case.
 Timothy Murphy email: gayleard /at/ eircom.net School of Mathematics, Trinity College, Dublin 2, Ireland

