Is there an eventually increasing sequence of natural numbers $(a_n)^\infty _{n=1}$ such that $\exists k \in \mathbb{N} : \lim_{n\to\infty} \frac{a_{n+k}}{a_n} \notin A $, where $ A $ denotes the set of algebraic numbers. I am expecting that this will always converge to some algebraic number yet I have failed to prove so. Can anyone provide some useful pointers on this?
1 Answer
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3 Let $$a_0=3, a_n=\lfloor \pi a_{n-1} \rfloor$$ You have $a_n \in \mathbb{N}$ and $\frac{a_{n+1}}{a_n} \to \pi$
EDIT : Proof :
$$\forall n \in \mathbb{N}, a_n \pi-1 \leq \lfloor \pi a_n \rfloor < a_n \pi+1$$ $$ \pi -\frac{1}{a_n} \leq \frac{a_{n+1}}{a_n} < \pi +\frac{1}{a_n}$$ and since $a_n \to \infty$ then$ \frac{a_{n+1}}{a_n} \to \pi$
- $\begingroup$ For a_0=3;and a_1=9; pi<=9/3=3 is false. $\endgroup$namen– namen2017-06-16 08:34:27 +00:00Commented Jun 16, 2017 at 8:34
- $\begingroup$ @namen sorry, I edited $\endgroup$stity– stity2017-06-16 08:42:21 +00:00Commented Jun 16, 2017 at 8:42
- $\begingroup$ accepted, thank you very much $\endgroup$namen– namen2017-06-16 08:43:00 +00:00Commented Jun 16, 2017 at 8:43