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Suppose $a_{n}$ is a sequence of real numbers then does $a_{n+1} = \frac{4}{a_{n}} + \frac{a_{n}}{2}$ converge?

I can find to where it converges, by setting $\lim a_{n+1} = a_{n} =x$ and then solving for $x$, but I don't know how to prove it's convergence.

A.M - G.M gives $$\frac{4}{a_{n}}+ \frac{a_{n}}{2} \geq 2 \cdot \sqrt{2}$$ but that doesn't seem to help :(

timur
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  • Show that the sequence is decreasing and bounded below, or increasing and bounded above (depending on positive or negative initial term). The sequence may not begin to be decreasing (resp increasing) until after the first few terms, depending on the value of $a_0$. – 2'5 9'2 Apr 15 '13 at 16:25
  • The solution has been given here: http://math.stackexchange.com/questions/359290/prove-the-converges-of-the-followin-sequence-and-find-the-limit/359302#359302 – Matt L. Apr 15 '13 at 16:26

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