<s>[INST] Convergence and Limit of a Recursive Sequence from the multiples of $a_n$ I'm having trouble with this recursive sequence problem. I'm supposed to find the limit, assuming that is it convergent, but I can't seem to get the answer.
$a_1 = 1, a_{n+1}= \frac {2a_n}{7+a_n}  $ 
 [/INST] We may just find the explicit form of $a_n$. By setting $a_n=\frac{p_n}{q_n}$ we have:
$$\begin{pmatrix} p_{n+1} \\ q_{n+1}\end{pmatrix}=\begin{pmatrix} 2 & 0 \\ 1 & 7\end{pmatrix}\begin{pmatrix} p_{n} \\ q_{n}\end{pmatrix}$$
where the characteristic polynomial of the involved matrix is $(x-2)(x-7)$.
By the Cayley-Hamilton theorem it follows that:
$$ p_n = A\cdot 7^n + B\cdot 2^n,\qquad q_n = C\cdot 7^n + D\cdot 2^n $$
and since $p_1=q_1=1$ and $p_2=2,q_2=8$ we have:
$$ p_n = 2^{n-1},\qquad q_n = \frac{6}{35}\cdot 7^n-\frac{1}{10}\cdot 2^n $$
so:

$$ a_n = \color{red}{\frac{5}{6\cdot \left(\frac{7}{2}\right)^{n-1}-1}}$$

and $\lim_{n\to +\infty} a_n = \color{red}{0}$ trivially follows.
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