In article <firstname.lastname@example.org>, email@example.com wrote:
> Den torsdagen den 1:e augusti 2013 kl. 23:30:58 UTC+2 skrev Virgil: > > In article <firstname.lastname@example.org>, > > > > email@example.com wrote: > > > > > > > > > The convergents of a continued fraction expansion of x give the best > > > > > rational approximations to x. > > > > > > > > But unless one of those convergents is EXACTLY equal to that x, > > > > that x is not itself rational. > > > > > > > > And being merely "close" does not count! > > > > -- > > If i show that a continued fraction converge to a limit, it doesn't mean that > i can use the number of vertices to find the limit but that i can show the > sequense is convergent.
Convergence of the convergents of a continued fraction does not establish rationality of the limit.
The test for rationality is whether the number in question is exactly equal to the quotient of two integers. If so it is rational, if not it is not.