
Re: Sequence limit
Posted:
Oct 26, 2013 3:13 AM


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.

Roland Franzius

