The most famous example of this is the Fibonacci sequence and the golden ratio.
I feel like this should be obvious. But, if I have an algebraic number say:
$\tan \left(\frac{15π}{16} \right) \approx -0.19891236738$
Which is the root of a irreducible 4th degree polynomial. Can I find a sequence where the ratio of consecutive terms approaches this number?
Can I always do this if I can do it some of the time?