I have a recursive string
$\displaystyle C_1 = 10, \quad C_{n+1}=4-\frac{4}{C_n}, \quad n>0.$
How can I find the limit, $\lim_{ n\rightarrow\infty}C_n$ of this recursive function?
Also, would like to determine (with proof) whether or not this sequence is monotone.
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First by solving $l=4-{4\over l}$ we obtain $l=2$. Second, define $$e_n=C_n-2$$therefore $$e_{n+1}=2-{4\over e_n+2}={2e_n\over e_n+2}$$Note that $$e_2=1.6$$and$$e_n<2\implies e_{n+1}<e_n$$also we cannot have $e_n>\epsilon$ for all $n>3$ and some $\epsilon>0$ since this leads to$$|e_{n+1}|\le {2\over 2+\epsilon}|e_n|$$ which leads to a contradiction. Hence $|e_n|\to 0$ and $C_n\to 2$.