karl
Posts:
228
Registered:
8/11/06


Re: Sequence limit
Posted:
Oct 26, 2013 3:51 AM


Am 26.10.2013 09:13, schrieb Roland Franzius: > Am 05.10.2013 09:04, schrieb quasi: >> Roland Franzius wrote: >>> quasi wrote: >>>> >>>> How do you justify the claim >>>> >>>> abs(sin(eps_j)) < (1/2)*eps_j >>>> >>>> ?? >>> >>> We called it the epsilon trick. >> >> Definitely quite a trick since, for 0 < eps_j < Pi/2, the above >> inequality is always false. > > > The conjecture that liminf_{n > oo} sin n^(1/n) = 1 seems more or less plausible from computer experiments. > > Using Mathematica, I just checked the first 50 continued fraction approximations of pi which give the series of best > rational bounds of pi within one unit of the denominator > > cf(pi,n) = a_n/b_n > > then > > log a_n is prop 1.15 n > > and > > > log(sin a_n) is prop  1.15 n > > and for n<50 > > 0.5 < a_n * sin a_n < 1.5 > > For n greater than 55 my system with 8 GB gives up. > > Of course these experiments do not rule out, that very good approximations of pi occur infinitely often, but to me it > seems more or less improbable that a subseries a_n_k of approximations giving > > liminf sin(a_n_k)^1/(a_n_k) < 1 > > may exist. > > > See more on > > http://mathworld.wolfram.com/PiContinuedFraction.html > > for the series q_n ^(1/n) for the denominators q_n with a seemingly constant limit which according to Eric Weisstein > has not yet proven. > > Here n is the index of the nth denominator in the truncated continued fraction series. After complete cancellation of > common factors, the denominators and numerators grow exponentially with the index. > > This is no answer to quasi's question, clever Dick!

