TITLE: A very curious rational fraction that converges. What is the value?
QUESTION [19 upvotes]: Is there any closed form for the following limit?

Define the sequence
  $$ \begin{cases} 
  a_{n+1} = b_n+2a_n + 14\\
  b_{n+1} = 9b_n+ 2a_n+70
\end{cases}$$
  with initial values $a_0 = b_0 = 1$. Then $\lim_{n\to\infty} \frac{a_n}{b_n} = ? $

The limit is approximately $0.1376$. My math teacher Carlos Ivorra says that this limit have a closed form involving the sine of an angle. What is the closed form for is limit?
NOTE: I have found this (and another series of converging sequences) by the use of an ancient method for calculating sines recently rediscovered. I'll give the details soon as a more general question.

REPLY [6 votes]: Here is a rigorous and systematic way to analyze the existence and value of the limit. (miracle173's answer didn't prove existence but uses essentially the same method for finding the value if it exists.)
Let $c_n = \frac{a_n}{b_n}$ (for each $n \in \mathbb{N}$).
Then $b_{n+1} c_{n+1} = b_n + 2 b_n c_n + 14$.
And $b_{n+1} = 9 b_n + 2 b_n c_n + 70$.
Thus $c_{n+1} = \dfrac{ b_n + 2 b_n c_n + 14 }{ 9 b_n + 2 b_n c_n + 70 } = \dfrac{ 1 + 2 c_n + \frac{14}{b_n} }{ 9 + 2 c_n + \frac{70}{b_n} }$.
Let $r = \dfrac{\sqrt{57}-7}{4}$ so that $r = \dfrac{1+2r}{9+2r}$, and let $d_n = c_n - r$.
Take any $ε > 0$, and let "$[x]$" denote "$\{ t : t \in \mathbb{R} \land |t| \le x \}$".
Then $c_{n+1} \in \dfrac{ 1 + 2 c_n }{ 9 + 2 c_n } + [ε]$ as $n \to \infty$, since $b_n \to \infty$ as $n \to \infty$.
Then $d_{n+1} \in \dfrac{ 1 + 2 (r+d_n) }{ 9 + 2 (r+d_n) } + [ε] - r = \dfrac{ 2 - 2 r }{ 9 + 2 (r+d_n) } d_n + [ε] \subseteq [\frac29 d_n] + [ε]$.
Thus $d_n \in [2ε]$ eventually as $n \to \infty$, since $\frac29 x + 1 < x - \frac12$ for any $x > 2$.
Therefore since $ε$ was arbitrary, $d_n \to 0$ as $n \to \infty$.